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'1. 10,  ivd. 


Inductiua  Sams, 


THE 

PRACTICAL 

AEITHMETIC 

ON  THE 

INDUCTIVE   PLAN, 

INCLUDING 

ORAL  AND  WRITTEN  EXERCISES. 

BV  % 

WILLIAM  J.  MILNE,  Ph.  D.,  LL.D., 

PBINCII'AL.  or  TUK  STATIi  NORMAL  SCHOOL,  OKNICSEU,   N.  Y. 


JONES  BROTHERS  &  COMPANY 

CINCINNATI,  CHICAGO, 

PHILADKLPHIA. 


.i^ 


Copyright,  1877,  by  John  T.  Joneb. 

SDOCATIOI  V5M% 


F.I.F.CTBOTYPED  AT 
IF.   FRANKLIN   TYPE  FOUNDRY, 
CINCINNATI. 


The  design  of  the  author  in  preparing  this  work  has  been 
to  embrace  within  moderate  compass  all  the  essentials  for 
V  Practical  Course  in  Arithmetic,  and  to  present  every 
subject  in  such  a  manner  as  to  secure  the  highest  mental 
development  of  the  learner.  To  accomplish  these  results  the 
author  has  spent  much  time  in  investigation,  and  in  consul- 
tation with  eminent  educators  and  successful  business  men, 
and  he  believes  that  he  has  included  in  this  volume  all  the 
subjects  necessary  for  the  arithmetical  part  of  a  business 
education. 

The  method  of  introducing  each  subject  is  such  that  the 
student  is  led  to  truth  in  the  path  of  the  original  investi- 
gator— certainly  the  most  natural  and  delightful  road  to  the 
acquisition  of  knowledge.  It  is  because  of  this  special  feat- 
ure in  connection  with  every  subject  that  the  series  has  been 
allied  The  Inductive  Series. 

The  work  contains  oral  and  written  exercises  sufficient  in 
number  to  enable  the  student  to  master  the  principles  un- 
derlying each  subject  and  to  give  him  fecility  in  numerical 
]>roce88e8. 


IV  PREFACE. 

In  the  problems  given  for  solution  it  has  been  the  aim  of 
the  author  to  use  the  language  of  trade,  when  no  error  is 
conveyed  thereby,  thus  accustoming  the  student  to  the  forms 
of  expression  needed  in  after  life ;  and  in  general  the  author 
has  striven  after  clearness  of  statement  rather  than  technical 
accuracy  of  expression. 

It  would  be  pedantry  to  specify  the  departments  in  which 
excellence  or  originality  may  be  found,  but  it  is  hoped  that 
a  careful  examination  will  exhibit  the  logical  sequence  of  the 
steps  in  all  the  processes,  the  perspicuity  and  accuracy  of  the 
analyses,  and  the  brevity  and  correctness  of  the  definitions, 
principles,  and  rules. 

The  author  takes  pleasure  in  acknowledging  his  indebted- 
ness to  Prof  J.  B.  De  Motte,  of  Indiana  Asbury  University, 
and  to  several  other  teachers  of  ability  and  experience,  for 
timely  and  valuable  suggestions. 

Trusting  that  the  book  will,  in  some  measure,  supply  the 
ix)pular  demand  for  a  brief  and  comprehensive  treatise  upon 
Arithmetic,  the  author  presents  his  work  to  the  public. 

W.  J.  M. 

8TATE  Normal  School, 
Geneseo,  a.  Y., 


-wa 

m 

]3^Sm 

Paok 

^ranqpm^ 

Pack 

Notation  and  Numeration     7 

Fractional  Forms  .     . 

122 

Arabic  Notation    .    . 

8 

Fractional    Relation    o 

Boman  Notation    .     . 

17 

Numbers  .... 

123 

Addition 

19 

Review  Exercises  .     . 

125 

Subtraction 

33 

Decimal  Fractions  ,    . 

129 

Multiplication    .    .    . 

43 

Reduction  of  Decimals 

134 

Division 

57 

Addition  of  Decimals 

138 

Analysis  and  Review. 

71 

Subtraction  of  Decimals 

139 

Properties  of  Numbers 

75 

Multiplic'n  of  Decimals 

141 

Divisibility  of  Numbers 

76 

Division  of  Decimals, 

143 

Factoring 

78 

Short  Processes .     .     . 

146 

Multiplicat'n  by  Factors 

79 

Accounts  and  Bills    . 

151 

Division  by  Factors    . 

80 

Review  Exercises  .    . 

155 

Cancellation  .... 

.      82 

Denominate  Nujibers  . 

157 

Common  Divisors  .    . 

84 

Measures  of  Value.    . 

158 

Multiples 

88 

Reduction  Descending 

160 

Fractions 

93 

Reduction  Ascending. 

162 

Reduction  of  Fractions 

96 

Measures  of  Space.    . 

164 

Addition  of  Fractions 

104 

Linear  Measures    .    . 

165 

Subtraction  of  Fractions 

107 

Surface  Measures  .    . 

167 

Multiplication   of   Frac- 

Measures of  Volume  . 

171 

tions     

.    109 

Board  Measure .    .    . 

174 

Division  of  Fractions. 

115 

Measures  of  Capacity. 

(V) 

176 

VI 


Measures  of  Weight 

Measures  of  Time  . 

Measures  of  Angles 

Beduction  of  Denominate 
Fractions  . 

Review  Exerci 

Addition  of  Denominate 
Numbers  .... 

Subtraction  of   Denomi 
nate  Numbers   .    . 

Multiplication     of     De- 
nominate Numbers 

Division  of  Denominate 
Numbers  . 


Longitude  and  Time  . 
Metric  System 
Percentage.    . 
Interest     .     . 
Method  by  Aliquot  Parts. 
Six  Per  Cent.  Method. 
Compound  Interest 
Annual  Interest     ,     . 
Partial  Payments  .     . 
Problems  in  Interest . 


CONTENTS. 

Pao« 

Pack 

178 

Commercial  Discount.     . 

237 

182 

True  Discount. .... 

i':?8 

134 

Bank  Discount  .... 

■J  111 

Be  view  Exercises  .     .     . 

•J45 

186 

Profit  and  Loss.     .     .     . 

247 

190 

Commifwion 

?r>4 

Beview  Exercises  .     .     . 

192 

Taxes 

202 

Duties 

?A^r^ 

194 

Stocks 

Insurance 

19G 

Exchange 

^•^t 

Average  of  Payments.    . 

290 

197 

Average  of  Accounts .    . 

295 

199 

Partnership 

298 

202 

Batio 

303 

207 

Proportion 

306 

218 

Involution 

315 

222 

Evolution 

319 

223 

Progressions 

336 

225 

Mensuration   .                 . 

341 

228 

Miscellaneous  Examples 

356 

229 

TJtsT  Questions    .... 

363 

232 

Answers 

372 

OTATI 0 N  &  NUM £R ATtOM 


TIvT 


Article  1.  A  Unit  is  a  single  thing. 

2.  A  Number  is  a  unit  or  collection  of  units. 

3.  In  counting  a  large  number  of  ol)joots  it  i«  natural  to 
group  them. 

Thus,  coins  are  put  in  piles  and  the  piles  counted,  envelopes  in 
packages  and  the  package*  counted,  etc.  These  piles  and  packages 
may  themselves  be  piled  and  put  in  larger  packages,  and  the  proccue 
continued  indefinitely. 

4.  To  express  numbers  so  that  all  may  understand  what 
is  meant  by  the  characters  which  represent  them,  the  system 
of  grouping  hy  lens  has  been  adopted.  There  are,  therefore, 
single  things,  or  unit»;  groups  of  ten  units,  or  tens;  groups 
containing  ten  tens,  or  hundreds^  etc. 

6.  The  method  of  grouping  hy  tern  is  called  the  Deci- 
mal System. 

Decimal  is  derived  from  the  Latin  word  decern,  wliich  means  in:. 


8  NOTATION   AND   NUMERATION. 

Numbers  may  be  expressed  by  loorda  or  other  eharaden, 
viz:  figures  and  letters. 

C  Notation  is  the  method  of  expressing  numbers  by 
figures  and  letters. 

The  Arable  Notation  Is  the  method  of  ezprcaslng  numbers  by  means  of 
Htfures.  lU  name  is  derived  from  Uie  Arabs,  by  whom  it  was  introduced  Into 
Europe.  The  Roman  Notation  is  the  method  of  expressing  numbers  by 
means  of  letters.    It  is  so  called  because  it  was  used  by  the  ancient  Romatus. 

7.  Numeration  is  the  method  of  reading  numbers 
expressed  by  figures  or  letters. 


ARABIC  SYSTEM. 

8.  In  this  system  ten  figures  are  employed  to  express 
numlx^rH,  viz: 

mgures.    0,    1,   2,    3,    4,    5,   6,    7,     8,    9. 

Names*     Naogbt,  Ooe,  Two,  Three,  Four,  Five,  Six,  Seven,  Eight,  Nine. 

Each  of  these,  except  naught,  is  called  a  tignifoamt  jigure. 
Naught  ifl  also  called  aero  and  c^^her. 

9.  By  combining  these  figures  according  to  certain  prin- 
ciples, we  can  express  any  number. 

10.  Principle.  —  Wfien  fibres  are  written  side  by  side,  the 
one  at  the  right  expresses  units,  the  next  tens,  and  the  next 
hundreds. 

BXBJtCI8B8, 

11.  1.  In  79,  what  does  the  7  express?  What  does  the 
9  express?     Read  the  number,  beginning  at  the  left. 

2.  In  58,  what  does  the  5  express?  What  does  the  8 
express?     Read  the  number,  beginning  at  the  left. 

3.  In  740,  what  does  the  7  express?  What  does  the  4 
express?     What  does  the  0  express?     Read  the  number. 

4.  Begin  at  the  left  and  read  76,  176,  106,  360,  203. 


NOTATION  AND   NUMERATION.  9 

12.  Figures  in  uniU^  place  express  units  of  the  first  order; 
those  ill  tens'  place,  units  of  Oie  second  order;  those  in  hundred^ 
place,  units  of  tlie  tlurd  orders  etc. 

13.  Nuniliers  between  1  ten  and  2  tens  are  named  thus: 

1  ten  and  1  unit    or  11,  eleven. 

1  ten  and  2  units  or  12,  twelve.' 

1  ten  and  3  units  or  13,  thirteen. 

1  ten  and  4  units  or  14,  fourteen. 

1  ten  and  5  units  or  15,  fifteen. 

1  ten  and  6  units  or  16,  sixteen. 

1  ten  and  7  units  or  17,  seventeen. 

1  ten  and  8  units  or  18,  eighteen. 

1  ten  and  9  units  or  19,  nineteen. 

Tlic  words  thirteen,  fourteen,  fifteen,  etc.,  mean  three  and  ten, /our 
and  ten,  five  and  ten,  etc. 

14.  The  units  of  die  second  order  are  named  as  follows: 


2  tens  or  20,  twenty. 

3  tens  or  30,  thirty. 

4  tens  or  40,  forty. 

5  tens  or  50,  fifty. 


6  tens  or  60,  sixty. 

7  tens  or  70,  seventy. 

8  tens  or  80,  eighty. 

9  tens  or  90,  ninety. 


The> suffix  ty  means  ten.    Thus,  forty  means  four  tetUj  etc. 

The  other  numbers  between  20  and  100  are  read  without 
the  word  and  between  the  tens  and  units.  Thus,  27  is  read 
twenty-seven,  instead  of  twenty  atui  seven. 


EXERCI8JE8. 


15.  litmd  the  folic 

)wing: 

28            99 

43 

73 

67 

41 

•  1}            83 

7.') 

86 

45 

31 

:iu           78 

m 

51 

32 

92 

47            32 

21 

55 

82 

25 

10 


NOTATION    AND    MMIKAPION. 


the  f-.ll.)uin<r: 

Five  tens  and  seven  units. 
Eight  tens  and  one  unit. 
Seven  tens  and  three  units. 

<  )ii«'  till  and  ci-lii  unils. 
J'" 1 11-  t»  ii>  and  two  units. 


1().    l']x])r('>,-  in  I'lLMii'*' 
Three  tens  and  v\'j:\\{  units. 
Four  tens  and  seven  units. 
Two  tens  and  two  units. 

( )nr  ten  and  ilirrc  units. 
{Six  tens  an(|  nine  units. 

Three  units  oi'  the  .-((mmhI  order,  >ix  ol'  the  first  order. 
Two  units  (1  tht  >((•.. ud  (.rder,  four  of  the  first  order. 
Write  all  th»    nuniUrs  l>etween  10  and  20.     Between  30 

and  50.     J^.  tv^   -•  To  <nd  'lO 

17.   Til  nuiiiim  u   imiiii^vi    I  \^»res>ed  l»v  m,"    iigures,  the 
It  n>    an     nad   after   the   hundreds  without    the  word  atid. 

Thus,  235   i>    r«a  1    tw..   hundrttl    thirty-five   instead   of   two 
hundred  a/j'/  thiit\-ti\t'. 


18. 


Head  the 

lollnW 

ng: 

746 

1)32 

786 

84? 

534 

678 

453 

777 

391 

585 

963 

378 

243 

873 

412 

531 

217 

918 

855 

248 

It).  I'x press  in  figures  the  following: 

Two  hundreds,  tluve  t*  n-.  live  units. 

Six  hundreds,  two  tens,  nine  units. 

Four  luindrod?.  one  ten,  eiLdit  units. 

Three  units  (»f  the  third  order,  six  of  the  second  order. 


Three  hundred  eighteen. 

Eight  liundred  tliirty. 
Four  hundred  four. 
Six  hundred  eiiihty-one. 
Seven  hundred  seventy. 
Seven  hundred. 


Seventy. 

Seven. 

Seven  hundred  six. 

Six  liundred  forty. 

Two  hundred  six. 

One  hundred  eleven. 


Seven  hundred  seventv-seven. 


NOTATION    AND   NUMERATION.  H 

From  the  previous  examples  we  deduce  the  following  gen- 
eral principles: 

"20,  Princii'livs. — 1.  The  rqnreserUative  value  of  a  figure  is 
increased  tenfold  by  each  removal  one  place  to  the  left,  and  de- 
creased tenfold  by  each  removal  one  place  to  the  right. 

2.  The  figure  0  is  used  to  give  significant  figures  tJieir  positiom. 

21.  In  reading  numbers  a  new  name  is  given  the  order  of 
units  next  higher  than  hundreds  of  any  denomination.  Thus, 
the  order  next  higher  than  hundreds  is  called  thousands, 
that  next  higher  than  hundreds  of  thousands,  millions,  etc. 
Therefore  each  denomination  can  have  but  three  orders  of 
units. 

22,  A  Period  is  a  group  of  figures  containing  the  hun- 
dreds, tens,  and  units  of  any  denomination. 

The  present  system  of  notation  is  illustrated  by  the  following 

TABLE. 
Periods.         6«/i.      5</i.       4th.       Zd.        2d.        1st. 


*F      I     I     §     i 


Namfs 
Periods. 


Orders. 


^§  ii  J  S  I  -^ 

<§         g  I  I  ^  P 


^  2  2  £  2  1 

WhPWh^WhPWehPWehPWehP 
3  0,2  3  0,1  6  0,7  0  0,4  0  1,6  9  0. 


This  number  is  read  tJiirty  quadrillion,  two  hundred  Hiirty 
trillion,  one  hundred  sixty  billion,  seven  hundred  million, /our 
hundred  one  thousand,  six  hundred  ninety. 


12 


KOTATION   AND   NUMERATION. 


1.  In  reading  numbers,  the  name  of  unito'period  is  omitted. 

2.  Each  period,  except  the  highest,  must  contain  three  figtues. 

3.  Tiie  periods  are  separated  from  each  other  by  commtu.^ 

28.  The  periods  above  Quadrillions,  iii  their  order,  are 
QuintiUionSt  SextUlions,  SepUUumg,  etc 

24.  Give  the  number  of  each  of  the  following  periods: 
Millions.  j  Trillions.  I       Billions. 
Thousands.        |          Units.               |       Quadrillions. 

25.  Give  the  names  of  the  following: 
5th  period.        I        2d  period. 
3d   period.        |        4th  period. 

26.  Repeat  in  order  the  names  of  the  periods  from 


Ist  period. 
6th  period. 


Units  to  billions. 
Units  to  quadrillions. 
Thousands  to  trillions. 


Billions  to  units. 
Quadrillions  to  units. 
Millions  to  thousands. 


27.  Copy  and  point  off  into  periods: 

1.  46825.  5.  38420058. 

2.  239746.  6.  33468204. 

3.  180040.  7.     8438206. 

4.  14168843.  8.  .    436784. 


9.       5284325684. 

10.  7932468512. 

11.  83749275867. 

12.  1423789276586. 


13.  How  many  thousands  are  there  in  the  first  number? 

14.  How  many  thousands  in  the  second  number? 

15.  How  many  billions  in  the  next  to  the  last  numlier? 

16.  How  many  trillions  in  the  last  number?  How  many 
billions?  How  many  millions?  How  many  thousands?  How 
many  units? 

17.  Point  off  into  periods,  and  name  in  order,  the  billions, 
millions,  thousands,  and  units  of  the  next  to  the  last  number. 

18.  Point  off  into  periods,  and  name  in  their  order,  the 
periods  composing  the  12th  number. 

19.  In  like  manner  point  off  and  read  each  of  the  numbers. 


34 


000 


018 


040 


NOTATION   AND    NUMERATION.  13 

'ZS,  Write  in  figures: 

1.  Thirty-four  billion,  eighteen  thousand,  forty. 

PROCESS  Analysis.— Since  the  highest  period  is  bill- 

ions, which  occupy  the  fourth  period,  we  make 
^  S  S  tS  ^<^"''  spaces  for  the  perio<is.  We  write  34  in  the 
fourth  period,  thus  expressing  the  billions  of 
the  given  number;  18  in  the  second  period,  thus 
expressing  the  thousands;  and  40  in  the  first 
OTj  period,  thus  expressing  the  units.     Since  every 

^4.  000  01 M  040     P^^''^^  except  the  highest  must  contain   three 

'        '        '  figures,  we  fill  the  vacant  places  with  ciphers. 

As  soon  as  pn^ssible  use  commas  instead  of  the  lines,  and  ceaae  to 
write  both  the  number  and  name  of  the  periods. 

Write  in  figures,  and  read  the  number: 

2.  Thirty-six  in  the  3d  period,  two  hundred  eighteen  in 
the  2d,  eight  hundred  forty-six  in  the  1st. 

3.  Eighty-four  in  the  4th  period,  five  hundred  forty  in 
the  3d,  six  hundred  in  the  2d,  forty  in  the  1st. 

4.  Two  hundred  one  in  the  5th  period,  seventy-five  in  the 
4th,  five  hundred  sixty-two  in  the  3d,  twelve  in  the  2d,  one 
in  the  1st. 

5.  Sixty  in  the  5th  period,  four  iiuudivd  two  in  the  4th, 
three  Imndred  thirty-three  in  the  3d,  two  hundred  in  the 
2d,  one  hundred  eleven  in  the  Ist. 

Write  in  figures: 

6.  Seventy-three  million,  two  hundred  fourteen  thousand, 
seventy. 

7.  Eighty  billion,  forty  million,  six  hundred  twelve  thou- 
sand, seven  hundred  eighty-eight. 

8.  Two  hundred  twenty-five  million,  six  hundred  forty- 
one  thousand,  three  hundred  fifly-one. 

9.  Three  hundred  fifty-four  billion,  six  hundred  four  mill- 
ion, eight  hundred  ninety-two  thousand,  thirty-six. 


u 


NOTATION    AND   NUMERATION. 


2\K  Rule  for  Notation.— fiegfin  at  the  l^  and  write  the 
hundreds^  tens,  and  units  of  each  period  in  tJieir  proper  order, 
pittting  ciphers  in  all  vacant  places  ajid  periods. 

While  writing,  separate  each  period  from  the  next  hij  a  comma. 

30.  Rule  for  Numeration. — Begin  at  tlic  right  ami  i^qj- 
arate  the  numbers  into  periods  of  three  figures  each. 

Begin  at  the  l^  hand  and  read  each  period  as  ^  it  stood 
alonCy  adding  its  name. 

BXERCISEa. 


31. 

Copy,  point  off,  and  read: 

1. 

116234. 

8.  141120. 

15. 

7640. 

2. 

65231. 

9.  101207. 

16. 

800900. 

3. 

20703. 

10.  68978. 

17. 

2568242. 

4. 

71005. 

11.  72020. 

18. 

1008003. 

5. 

3104. 

12.  80001. 

19. 

212375647 

6. 

48000. 

13.  857(kH> 

20. 

609003588 

7. 

60029. 

14.  91029. 

21. 

897856846 

32.  Write  in  Bgures,  and  read: 

22.  Two  hundred  in  the  Ist  period. 

23.  Sixty  in  the  2d  period,  two  in  the  Ist. 

24.  Seven  hundred  in  the  3d  period. 

25.  Two  hundred  thirty  in  the  3d  period,  sixty  in  the  Ist. 

26.  Eighty-one  in  the  4th  period,  fivt-  lumdnMl  on*'  in  tlie 
3d,  seven  in  the  2d,  twelve  in  the  1  - 

27.  Thirty  in  the  5th  period,  six  hundred  three  in  the  Ist. 

28.  Seven  hundred  in  the  5th  period,  eighty  in  the  4th. 

29.  Eight  in  the  4th  period,  seven  in  the  3d,  fourteen  in 
the  2d,  and  ten  in  the  1st. 

30.  Fifteen  in  the  6th  period,  eighteen  in  the  4th,  two 
hundred  seven  in  the  3d,  and  eighty-one  in  the  Ist. 


>'OTATION    AND    NUMERATION.  13 


33.  Copy,  point  off,  and  read: 

1.  60701892.  7.  163194568 

2.  50607801. 

3.  600000. 

4.  49000000.  . 

5.  593006070500. 

6.  19019000190019019. 


8.  3050050183. 

9.  5000204. 

10.  594900. 

11.  12000012. 

12.  200798013400019. 


i:].  212506067093012063067. 

34,  Write  in  figures: 

14.  Two  in  the  3d  period,  sixty  in  the  2d,  one  hundred 
fifty-three  in  the  1st. 

15.  Sixty  in  each  of  the  4th,  3d,  2d,  and  1st  periods. 

16.  60  million,  200  thousand,  500. 

17.  402  billion,  348  million,  213  thousand,  20. 

18.  78  trillion,  640  billion,  9  million,  6  thousand,  16. 

19.  6  billion,  542  million,  25. 

20.  Six  billion,  five  hundred  forty-two  million,  twenty-five. 

21.  Four  hundred  two  billion,  three  hundred  forty-eight 
million,  two  hundred  thirteen  thousand,  twenty. 

22.  Five  million,  two  hundred  sixty-eight  thousaml,  nine 
hundrefl  forty-nine. 

23.  Two  hundred  million,  three  hundred  thousand,  eight 
hundred. 

24.  Twenty-nine  billion,  five  hundred  ninoty-nino  million, 
six  hundred  one. 

25.  Four   trillion,    five    hundred    fifty -eight    million,    t\v(» 
hundred  forty-four  thousand,  seventy. 

26.  Thirty-two  billion,  sixty-one   million,   three   hundred 
forty-three  thousand,  four  hundred  four. 

27.  Five  hundred  fifty-five  million,  seven  hundred  seventy- 
seven  thousand,  six  hundred  sixty-nine. 

28.  Eight  hundred  six  billion,  seventy  million,  three  hun- 
dred eighty-five  thousand,  two  hundred  six. 


16  NOTATION  AND  NUMERATION. 

29.  Nine  hundred  forty-one  trillion,  one  hundred  sixteen 
thousand,  twenty-two. 

30.  Twenty-three  billion,  twenty-three  million,  twenty- 
three  thousand,  twenty-three. 

31.  Six  hundred  thousand,  seventy-five. 

32.  Twelve  billion,  eight  million,  nine  hundred  eighty- 
eight  thousand,  thirteen. 

33.  Twenty-nine  quadrillion,  seven  hundred  fifty-seven  trill- 
ion, four  hundred  eighty  million,  thirteen  thousand,  five 
hundred  sixty-five. 


NOTATION  AND  NUMERATION  OF  U.  8.  MONEY. 

35.  The  curreney  of  the  United   States  has  a  Decimal 
System  of  notation,  thus: 

10  mills  make  1  cent 
10  cents  make  1  dime. 
10  dimes  make  1  dollar. 

M.  The  Sign  of  dollars  is  $.     It  is  written  before 
the  number. 

Thus,  $16  is  read,  sixteen  dollars. 

37.  In  writing  decimal  currency  a  mark  called  the  deci- 
mal point  is  placed  before  cents  and  mills. 

38.  Cents  occupy  the  fird  two  places  at  the  right  of  the 
decimal  point,  and  rniUs  the  third. 

Thus,   $7,584  is  read,  seven  dollars,  fifty-eight  cents,  four  mills; 
$.694  is  read,  sixty-nine  cents,  four  mills. 

39.  If  the  number  of  cents  is  less  tiian  ten,  write  a  cipher 
in  the  first  place  at  the  right  of  the  decimal  point. 

Thus,  five  dollars,  eight  cents,  Lj  written,  $5.08;  three  dollars,  seven 
tents,  $3.07. 


NOTATION    AND    NUMERATION. 

,  Read  the  following: 

«6.85 

87.843 

$12,056 

$31,095 

824.055 

$20.20 

$28,075 

$40.04 

$606,952 

$500.50 

$2103.094 

$7000.16 

$20000. 

$6001.101 

$300,416 

$212012.12 

$695,955 

$200,204 

$613,495 

$211.12 

$69.69 

$203,033 

$216.16 

$76.25 

17 


41.  Write  the  following: 

1.  Two  dollars,  twenty-three  cente,  five  mills. 

2.  Two  hundred  two  dollars,  two  cents,  five  mills. 

3.  One  hundred  twelve  dollars,  twenty-five  cents. 

4.  Six  hundred  two  dollars,  nine  cents. 

5.  Twenty  thousand  dollars,  thirty-two  cents. 

6.  Twelve  million,  seven  hundred  thousand  dollars. 

7.  Six  million  dollars,  eighty-eight  cents. 

8.  Twelve  thousand  three  hundred  dollars,  fifteen  cents. 

ROMAN   SYSTEM. 


42.  In  this  system  seven  letters  are  used  to  express  num- 
bers, viz: 

Letters.     I,    V,    X,     L,      C,       D,       M. 
Values.      1,    5,     10,     50,     100,    500,    1000. 

By  combining  these  letters  according  to  certain  principles 
any  number  can  be  expressed. 

Principles. — 1.   When  a  letter  it  repeated  Us  value  is  re- 
peated. 

Thus,   1   rcprt-rttntrt   1;  II,  two;  III,   three;  X,  ten;  XX,  twenty; 
XXX,  thirty  ;  C,  one  hundred;  CCC,  three  hundred. 


18 


NOTATION    AND   NUMERATION. 


2.  When  a  letter  is  plaeed  before  atwOier  of  greater  valve  Ua 
value  is  to  be  taken  from  that  of  the  greater. 

Thus,  I  roprt'sents  one  and  V  five,  but  IV  represents  four;  IX, 
nine;  XIX,  nineteen;  XL,  forty;  XC,  ninety. 

3.  When  a  letter  is  plaeed  after  another  (f  greater  value  their 
values  are  to  be  united, 

T1iu«,  XV  repref«ents  fifteen;  LX,  sixty;    i   \  \  \.  .  Lhty;  DC,  six 
hunrircNl;  MD,  fifteen  hundred. 

1  plaeed  over  a  number  increases  its  value  a  thmwind- 

fold. 

Thus,  V  represents  five;    V,  five  thousan  '      '  ^  1  X.    i\ty 

thousand;   M,  one  thousand;    M,  one  milli< 


TABLE. 

I  . 

11 

III 

IV       . 
V 

1 
2 

W 

4 

, 

6 

7 

8 

1) 

10 

11 

12 

13 

XIV 

XV 

XV  i 

XVIi.  .  . 
XVTTT 
XI  \ 
XX 
XX 

XXIX  .  . 

XXX  .  . 
XXXIV  . 
XL  .     .  . 
L 

.  .  .  11 

.  15 

.10 

.  .  .  17 

.  .  18 

.  19 

.  20 

.  .  21 

.  .  .  29 

.  .  .  30 

...  34 

.  .  .  40 

.60 

LX. 
LXX  .  . 
LXXX. 
XC  . 
C 

cx: 

CCL 
CCC 
D 

....     »;o 

-') 

.       1.0 

100 

VI 

200 

VII 

2.>0 

VII I 

.     400 

IX    .  . 

.     500 

X  .  . 

XI  . 

XII  .  . 

XIII  . 

DCC 700 

M        UKK) 

MMM 3000 

MDCCCLXXX  1880 

Read  the  followin 

g  numbers: 

XV;     XXIV; 
CCCLXXXIX; 

X> 

DC( 

:XIX;     XL; 
;XXXVI: 

XLIX; 
VDLV; 

XCIX; 
DLDC; 

LXXVII ; 
CCXDVI ; 

LXXMMMDCCCXCIX ;    MDXCVDCCCLXIV. 

Express  the  following  numbers  by  Roman  Notation: 
15,  18,  27,  81,  95,  86,  5J4,  684,  1050,  8004,  7000,  75869. 


r^ 


3:^^ 


♦-ADDITION 


IXnUCTIVE    e'xERCI8E8. 


43.    1.  How  many  are  2  pears  and  1  pear?    2  pears  and 
2  pears? 

2.  How  many  are  3  leaves  and  2  leaves?  3  leaves  and  3 
leaves?     How  many  are  3  and  1?     3  and  2?     3  and  3? 

3.  Jane  has  3  apples  and  Mary  has  4  apples.  How  many 
apples  have  both?    How  many. are  3  and  4?     4  and  3? 

4.  George  gave  me  2  apples  and  Mary  gave  me  4.  How 
many  apples  did  both  give  me?  How  many  are  4  and  2? 
2  and  4? 

5.  A  farmer  had  2  horses  and  bought  6  more.  How 
many  horses  had  he  then?  How  many  are  2  and  6?  6 
and   2? 

6.  Henry  paid  5  cents  for  a  pencil  and  7  cents  for  a 
writing-book.  How  many  cents  did  he  pay  for  both?  How 
many  are  5  and  7?     7  and  5? 

7.  If  a  barrel  of  flour  is  worth  S6,  and  a  cord  of  wood 
$4,  how  much  are  both  worth?     How  many  are  6  and  4? 

8.  A  man  plowed  8  acres  of  land  in  one  week  and  6  acres 
the  next  week.     How  many  acres  did  he  plow  in  both  weeks? 

9.  On  the  Fourth  of  July,  Ned  spent  10  cents  for  fire- 
crackers and  6  cents  for  torpedoes.  How  many  cents  did  he 
spend  for  both? 

10.  Harry  is  6  years  old  and  his  sister  is  tour  years  older. 
How  old  is  his  sister?     How  many  are  6  uikI  4? 


20  ADDITION. 

11.  At  Christmas,  Horace  received  9  gifts  from  his  par- 
ents, and  4  from  other  friends.  How  many  gifts  did  he 
receive? 

12.  A  certain  house  has  5  windows  in  one  side  and  7  in 
another,     il  >\\   many  windows  in  the  two  sides? 

13.  How  many  are  5  oranges  and  4  oranges?  6.  boys 
and  3  boys?     5  horses  and  G  cents? 

14.  Why  can  you  not  tell  how  many  5  horses  and  6  cents 
are?  ^ 

15.  Why  can  you  tell  how  many  5  oranges  and  4  or- 
anges are? 

Numbers  that  express  things  of  the  same  name  are  called 
Itike  NinNberH, 

16.  What  kind  of  numbers  only  can  be  united? 

DEFmrrioNS. 

44.  Addition  is  the  process  of  finding  a  niiml^cr 
which  shall  be  equal  to  two  or  more  given  numlxrs. 

45.  The  Sum  or  Amount  is  the  result  obtained  by 
adding. 

46.  The  Siffn  of  Addition  is  an  upright  cross:  -f-. 
It  is  called  pluSf  and  is  placed  between  numbers  to  be 
added. 

ThuB,  3  -f  4  is  read  3  plus  4,  and  means  that  3  and  4  are  to  be 
added. 

47.  T^e  Sign  of  Equality  is  two  short  horizontal 
lines:   =.     It  is  read  equals,  or  is  equal  to. 

Thus,  3  -|-  4  =  7,  is  read  3  plus  4  equals  7. 

Tlie  expression  3  -f  4  =  7,  or  any  other  expression  of 
equality,  is  called  an  Equation, 


ADDITION. 


21 


48.  Principles. — 1.   Otdy  like  numbers  can  be  added. 
2.   The  mm  and  numbers  added  mud  be  like  numbers. 

TABLE. 


1-fl 
2  +  1 
3-M 

4+1: 

5+1 
6+1 

7  +  1 
8+1 
9  +  1 


2 
3 
4 
5 
6 
7 
8 
9 
10 


1  +  2  = 

2  +  2: 

3  +  2: 

4+2: 

5  +  2: 

6  +  2^ 

7  +  2: 

8  +  2: 

9  +  2: 


3 
4 
5 
6 
7 
8 
9 
10 
11 


1  +  3: 

2  +  3: 

3  +  3: 

4+3: 
5+3: 

6  +  3: 

7  +  3: 

8  +  3: 

9  +  3 


4 
5 
6 

7 

§ 

9 

10 

11 

12 


1  +  4: 

2  +  4: 

3  +  4: 
4+4: 

5+4: 

6  +  4. 

7  +  4: 
8+4: 
9  +  4: 


6 

6 

7 

8 

9 

10 

11 

12 

13 


1+  5=  6 

2+  5=  7 

3+  5=  8 

4+  5=  9 

5+  5=10 

6+  5=11 

7+  5=12 

8+  5=13 

9+  5=14 


1  +  6: 

2  +  6: 

3  +  6: 

4+6: 
5+6: 
6+6: 

7  +  6 

8+6: 
9  +  6: 


:  7 

8 

9 

10 

11 

12 
13 
14 
15 


1  +  7: 

2  +  7: 

3  +  7: 
4+7: 
5  +  7: 
6+7: 

7  +  7: 

8  +  7: 

9  +  7: 


8 
9 
10 
11 
12 
13 
14 
15 
16 


1  +  8: 

2  +  8: 

3  +  8: 
4+8: 

5  +  8: 

6  +  8: 

7  +  8: 

8  +  8: 

9  +  8: 


9 
10 
11 
12 
13 
14 
15 
16 
17 


1  +  9: 

2  +  9: 

3  +  9: 
4+9: 

5  +  9: 

6  +  9: 

7  +  9: 

8  +  9: 

9  +  9: 


10 

11 

12 
13 
14 
15 
16 
17 
18 


1  +  10=11 

2  +  10=12 

3  +  10=13 
4+10=14 
5  +  10=15 
6+10=16 
7+10=17 

8  +  10=18 

9  +  10=19 


CASE    I. 
49.  To  add  siugle  colnniiis. 

1.  A  grocer  sold  8  pounds  of  sugar  to  one  man  and  7  pounds 
to  another*     How  many  pounds  did  he  sell  to  both  ? 

Analysis. — Since  he  sold  8  pounds  to  one  man  and  7  pounds  to 
another,  to  both  he  sold  the  sura  of  8  pounds  and  7  pounds,  which 
is  15  pounds. 

2.  A  man  rode  7  miles  the  first  hour  and  6  miles  the  sec- 
ond hour.     How  far  did  he  ride  in  the  two  hours? 


22  ADDITION. 

3.  On  one  tree  are  8  birds,  and  on  unothor  4  Wmh,     How 
many  birds  are  there  on  both? 

4.  Carl  earned  $2  in  May,  §5  in  Jiino,  and  >1  in  July. 
How  much  did  lie  earn  in  the  three  months? 

5.  I  gave  6  nuts  to  one  boy,  5  to  another,  and      i     an- 
other.    How  many  nuts  did  I  give  to  all? 

6.  I  paid  5  cents  for  paper,  3  cents  for  j)ens.  and  5  cents 
for  ink.     How  much  did  I  pay  for  all? 

7.  A  lemon  cost  5  cents,  an  orange  0  ccntis,  and  a  pine- 
apple 8  cents.     What  did  they  all  cost? 

8.  Esther  gave  her  teacher  5  pinks,  7  roses,  and  4  i)an- 
8168.     How  many  flowers  did  she  give  her? 

9.  James  shot  9  birds,   Henry  shot  6,  and  William   o. 
How  many  did  they  all  shoot? 

10.  A  woman  picked  9  quarts  of  blackberries  one  morn- 
ing, while  her  son  picked  3  quarts.  How  many  quarts  did 
both  pick? 

11.  James  solved  6  examples,  John  5,  William  8,  and 
Henry  7.     How  many  examples  did  they  solve  in  all  ? 

12.  One  boy  picked  6  quarts  of  cherries,  another  4  quarts, 
another  5  quarts.     How  many  quarts  did  they  all  pick? 

13.  I  gathered  from  one  pear-tree,  this  year,  2  bushels  of 
fruit,  from  another  4  bushels,  from  another  3  bushels,  and 
from  another  2  bushels.  How  many  busho]>  did  I  gather 
from  these  four  trees? 

14.  A  merchant  sold  from  a  piece  of  cloth,  3  yards  at  one 
time,  6  yards  at  another,  8  yards  at  another,  and  5  yards  at 
another.     How  many  yards  did  he  sell  in  all? 

15.  A  man  picked  8  barrels  of  apples  on  Monday,  6  bar- 
rels on  Tuesday,  4  barrels  on  Wednesday,  and  5  Imrrels  on 
Thursday.     How  many  did  he  pick  altogether? 

16.  Henry  learned  7  verses  of  poetry  on  one  day,  5  on 
another,  6  on  another,  and  8  on  another.  How  many  verses 
did  he  learn  in  the  four  days? 


ADJ'iiiuN.  23 

17.  A  man  jniid  $9  for  a  coat,  84  for  pants,  and  $2  for  a 
hat.     How  much  did  he  pay  for  all? 

18.  In  a  garden  there  are  8  apple-trees,  7  plum-trees,  and 
9  peach-trees.     How  many  trees  are  there  in  the  garden? 

19.  There  are  4  boys  and  7  girls  in  one  class,  and  6  boys 
and  8  girls  in  another.     How  many  pupils  in  Ixith  classes?  ^ 

20.  Homer  paid  8  dollars  for  a  fur  cap,  and  5  dollars  for 
a  pair  of  skates.     How  much  did  both  cost  him? 

21.  A  boy  gathered  nuts  for  three  days.  The  first  day  he 
brought  home  8  quarts,  the  next  day  7  quarts,  the  next  day 
9  quarts.     How  many  quarts  did  he  bring  home? 

22.  Repeat  the  addition  table  of  ones.  Of  twos.  Of 
threes.  Of  fours.  Of  fives.  Of  sixes.  Of  sevens.  Of 
eights.     Of  nines.     Of  tens. 

23.  Count  by  2's  from  0  to  20;  thus:  0,  2,  4,  6,  8,  10, 
12,  etc. 

24.  Count  by  3's  from  2  to  26. 

25.  Count  by  4*s  from  0  to  36. 

26.  Count  by  5's  from  3  to  43. 

27.  Count  by  6's  from  0  to  42. 

28.  Count  by  7's  from  4  to  39. 

29.  Count  by  8's  from  2  to  58. 

30.  Count  by  9's  from  7  to  70. 


WRITTJSN    EXERCISES. 

50.    1.  What  is  the  sum  <  i   'k    1.  7,  aul  t..^ 

PROCESS.  Analysis. — We  write  the  numbers  to  be  added  in  a 

5  column,  and  begin  at  the  bottom  to  add;  thus:  G,  13,  17, 

A  22;  and  write  the  sum  beneath.     To  prove  the  work  we 

may  begin  at  the  top  and  add  downwards.  If  the  rtsult 
agrees  with  the  one  formerly  obtainod  the  work  is  proba- 
bly correct.     In  adding  say,  (5,  i:J,  17,  etc.,  instm*!  -f  c, 


From  26  to  41. 

From 

5  to  53. 

From 

7  to  72. 

From 

4  to  46. 

From 

11  to  60. 

From 

7  to  63. 

From 

8  to  71. 

7 
6 
22    Slim,      and  7  are  1.3  and  4  are  17,  etc. 


24 


ADDITION. 


Copy,  add,  and  prove: 


(2.) 
5 
3 
4 
2 


(3.) 
6 
7 
8 
1 


(4.) 
5 

(> 
2 
3 


cr^^ 


(c^^ 

8 


(7.) 
8 
9 
8 
7 


(8.) 

(9.) 

.  ]t\  ^ 

1    ! 

'  \2.) 

(  13.) 

8 

6 

;i 

2 

7 

4 

■\ 

t 

7 

5 

6 

3 

•) 

u 

8 

7 

3 

0 

2 

s 

9 

9 

5 

3 

1 

8 

8 

3 

51.  Required  the  sum  of  the  foUowing: 


14.  6,  7,  5,  3,  2,  4,  and  5. 

15.  8,  2,  0,  3,  3,  2,  and  4. 

16.  5,  6,  7,  6,  4,  2,  and  8. 

17.  3,  2,  6,  5,  8,  7,  and  9. 

18.  8,  3,  0,  5,  3,  8,  and  2. 

19.  7,  6,  6,  4,  3,  6,  and  3. 


20.  7,  8,  8,  9,  0,  3,  and  3. 

21.  8,  9,  7,  8,  5,  8,  and  2. 

22.  7,  6,  5,  4,  3,  2,  and  1. 

23.  5,  4,  4,  3,  2,  6,  and  7. 

24.  4,  3,  4,  5,  6,  8,  and  8. 

25.  3,  6,  8,  6,  7,  0,  and  5. 


26.  There  are  8  chickens  in  one  coop,  9  in  anotht*r,  7  in 
another,  and  5  in  another.  How  many  chickens  are  there 
in  all  the  coope? 

27.  My  father  has  5  horses,  9  cows,  7  sheep,  and  3  pigs. 
How  many  animals  has  he  in  all? 

28.  A  man  walked  from  A  to  B  in  four  hours.  He  went 
4  miles  the  first  hour,  3  miles  the  second  hour,  5  miles  the 
third  hour,  and  6  miles  the  fourth  hour.  What  was  the 
distance  between  the  two  places? 

29.  A  house  had  8  windows  on  the  east  side,  7  on  the 
west,  and  9  on  the  soiith.     How  many  were  there  in  all? 


ADDITION.  26 

CASE    II. 
53.  To  add  several  colunius. 

1.  Count  by  Ws  from  7  to  107;  thus,  7,  17,  27,  37, 
47,  etc. 

2.  Count  by  lO's  from  5  to  95.     From  9  to  79. 

3.  Count  by  20's  from  5  to  85.     From  9  to  89. 

4.  Add  2  to  each  of  the  series  of  numbers  6,  16,  26,  etc., 
to  76. 

5.  Add  3  to  each  of  the  series  of  numbers  from  8,  18, 
etc.,  to  88. 

6.  A  gentleman  paid  $7  for  a  hat,  88  for  a  vest,  and  $13 
for  pantaloons.     How  much  did  he  pay  for  all? 

Analysis.— Since  he  paid  $7  for  a  hat,  $8  for  a  vest,  and  $13  for 
pantaloons,  for  all  he  paid  the  sum  of  $7,  $8,  and  $13,  or  $28. 

7.  James  gave  25  cents  to  his  brother  and  20  cents  to  his 
sister.  How  much  did  he  give  to  both?  25  and  20  are  how 
many? 

8.  Horace  earned  35  cents  on  Monday,  20  cents  on  Tues- 
day, and  9  cents  on  Wednesday.  How  much  did  he  earn 
during  the  three  days?     How  many  are  35  and  29? 

9.  William  saw  two  flocks  of  wild  geese;  the  first  of  27 
geese,  the  second  of  23  (20-|-3).  How  many  geese  did  he 
see?    How  many  are  27  and  23? 

10.  Paid  9  cents  for  raisins,  15  cents  for  plums,  and  27 
(20  -|-  7)  cents  for  currants.     How  much  did  all  cost? 

11.  During  a  certain  recitation  29  questions  were  answered 
correctly  and  16  incorrectly.  How  many  questions  were 
asked?    How  many  are  29  and  10?    39  and  6? 

12.  Add  2  to  each  of  the  numbers  2,  12,  22,  32,  42,  etc., 
to  72. 

13.  Add  3  to  each  of  the  numbers  4,  1  i,  21,  etc.,  to  94. 

14.  Add  4  to  each  of  the  numbers  9,  19,  29,  39,  to  99. 


26  ADDITION. 

IT).  Add  each  of  the  numbers  5,  i,  .,  ,  and  U  tu  each  of 
tlie  numl)ers  6,  16,  26,  etc.,  to  96. 

16.  A  certain  school  had  40  girls  and  30  l)oys  in  attend- 
ance.    How  many  pupils  were  there  in  the  school? 

17.  A  music  teacher  paid  $12  for  a  metronome  and  $15 
for  music.     How  much  did  she  jwy  for  both? 

18.  A  boy  bought  a  veloci|)ede  for  315  and  a  watch  for 
$20.     How  much  did  both  cost  him? 

19.  Mary  read  20  pages  of  history  one  day,  30  pages  the 
noxt,  and  25  the  next  How  many  pages  did  she  read 
ill  all? 

20.  In  a  certain  l)ook-ca5e  there  were  18  books  on  the 
upi)er  shelf,  20  on  the  next,  12  on  the  next,  and  10  on  the 
]<>W(  St.     How  many  books  in  the  case? 

21.  A  merchant  sold  15  yards  of  cloth  to  one  woman,  25 
to  another,  30  to  another,  and  25  to  another.  How  many 
yards  did  he  sell  to  them  all? 

22.  A  ix>stma8ter  sold  on  one  day  50  three-cent  stamps, 
65  on  another,  and  55  on  another.  How  many  stamps  did 
he  sell  in  the  three  days? 

23.  James  solved  31  oral  problems  and  24  written  prob- 
lems. Harry  solved  35  oral  problems  and  25  written  prob- 
lems. How  many  problems  did  each  solve?  How  many  did 
both  solve? 

24.  In  an  orchard  there  are  26  cherry-trees  and  31  apple- 
trees.     How^  many  trees  are  there  in  the  orchard? 

25.  Henr}'  saw  three  flocks  of  wild  ducks,  the  first  con- 
taining 17  ducks,  the  second  25,  and  the  third  30.  How 
many  ducks  did  he  see? 

26.  James  paid  28  cents  for  a  slate,  20  cents  for  a  writing- 
book,  and  10  cents  for  ink.     How  much  did  he  pay  for  all? 

27.  The  month  of  January  has  31  days,  the  month  of  Feb- 
ruary has  28  days,  and  the  month  of  March  has  31  days. 
How  many  days  are  there  in  these  three  months? 


ADDITION.  27 

28.  How  many  acres  are  there  in  three  fields,  containing 
resiKXjtively  22  acres,  33  acres,  and  37  acres? 

WRITTEN    EXBRCISES. 

63.    1.  What  is  the  sum  of  8535,  $213,  and  S384? 

PROCESS.  Analysis. — For  convenience  we  arrange  the  numbers  to 

$535  ^  adde<l  so  that  units  of  the  same  order  shall  stand  in  the 

910  sjime  column.    Beginning  with  the  lowest  order  of  unita 

„^  .  we  add  each  column  separately.    Thus,  4  4-34-5  =  12, 

the  sum  of  the  units.     12  units  are  equal  to  1  ten  and  2 


$1132  units.  We  write  2  under  the  column  of  units  and  reserve 
the  1  to  add  witli  the  tons. 

1  reserved  4- 8  4- 1  4- 3  =  13,  the  sum  of  the  tens,  13  tens  are 
equal  to  1  hundred  and  3  tens.  We  write  the  3  under  the  column 
of  teas  and  reserve  the  1  to  add  with  the  hundreds. 

1  reserved  4- 3  4- 2  4- 5  =  11»  the  sum  of  the  hundreds.  11  hun- 
dreds are  equal  to  1  thousand  and  1  hundred,  which  we  write  in 
thousands'  and  hundreds'  places  in  the  sum. 

Hence  the  sum  is  $1132. 

1.  In  adding,  name  reeults  only.  Thus,  instead  of  saying  4  and  3 
are  7  and  5  are  12,  say  4 ,  7,  12. 

2.  When  the  sum  of  any  column  is  exactly  10,  20,  or  any  number 
of  lem,  we  write  0  under  the  column  added  and  reserve  the  1,  2,  3, 
etc.,  to  add  with  the  next  column. 

54.  Rule. — Arrange  tiie  nunibers  so  iJwt  units  of  the  same 
order  sliall  stand  in  Vie  same  column. 

Begin  at  the  right ,  add  eacli  column  separately,  and  vn-He  Uie 
sum,  if  it  is  less  tJian  ten,  under  the  column  added. 

If  tJie  stnn  of  any  column  be  ten  or  more,  write  tJie  unit  figure 
only  wider  tliat  column  and  add  Vie  ten  or  tens  unth  Uie  next 
column. 

Write  die  entire  sum  of  Vie  last  column. 

Proof. — Add  each  column  in  tiie  reverse  order.  If  tJie  re- 
iiultji  agreCf  tJie  uxjrk  is  j/robably  correct. 


i 

ADDITION. 

EXAMPLES. 

Copy,  add, 

and 

prove: 

(2.) 

(3.) 

(4.) 

(5.) 

310 

612 

$24.15 

$12.25 

114 

415 

10.21 

9.08 

523 

371 

8.34 

7.15 

(6.)      (7.)     (8.)      (9.) 


4134 

8104 

3910 

45 

2460 

3673 

418 

3061 

3782 

1856 

1916 

418 

469 

7206 

39 

6 

10.  Add  4834,  3910,  4826,  8404. 

11.  Add  3159,  7816,  5459,  3568. 

12.  Add  $16.05,  $10.38,  $77,055. 

13.  Add  $317.50,  $600.10,  $514,085,  $6.10. 

14.  What  is  the  sum  of  thirty-six  thousand,  three  hundred 
five;  eight  hundred  ninety-seven  thousand,  nineteen? 

15.  What  is  the  sum  of  fifly-uine  thousand;  seven  thou- 
sand, three  hundred  twelve;  sixty-eight  thousand,  four  hun- 
dred twenty-seven? 

16.  What  is  the  sum  of  three  hundred  forty-four  million, 
'  iirht  hundred  ninety-six  thousand,  four  hundred  thirty-five ; 
live  million,  six  thousand,  three;  forty-eight  thousand,  two 
hundred? 

17.  What  is  the  sum  of  eighteen  dollars,  five  cents;  fifty- 
one  dollars,  fifty-one  cents;  ten  dollars,  ten  cents;  eighteen 
dollars,  twenty-four  cents*;  thirty-five  dollars,  four  cents? 

18.  A  owns  345  sheep,  B  owns  295,  C  owns  436,  D  owns 
)24l     How  many  sheep  do  all  own? 


ADDITION.  29 

19.  A  man  boKI  liis  ])iano  for  $413,  his  collection  of  jniint- 
ings  for  $536,  his  lil)rary  for  $719,  his  cari)ets  for  $728,  other 
furniture  for  $1,730,  his  horses,  carriage  and  two  sets  of  har- 
ness for  $1,324,  and  his  house  for  $9,137.  How  much  money 
did  he  obtain  by  the  sale? 

20.  A  fruitnlealer  shipped  for  New  York,  3,932  bushels 
of  apples  in  one  week,  2,436  in  the  next,  4,197  in  the  third, 
and,  within  the  next  month,  10,937  bushels.  What  was 
the  entire  number  of  bushels  shipixjd  by  him  during  that 
time? 

21.  A  man  making  his  will,  left  $3,450  to  his  wife,  $2,675 
to  his  oldest  son,  $1,850  to  his  second  son,  and  $1,290  to  his 
youngest  son.  What  amount  of  money  was  bequeathed  in 
his  will? 

22.  A  man  owns  five  horses.  The  first  is  worth  $225,  the 
second  $325,  the  third  $450,  the  fourth  as  much  as  the 
second  and  third,  and  the  fifth  as  much  as  the  first  and 
fourth.     What  is  the  value  of  the  five  horses? 

23.  A  and  B  were  building  a  brick  store.  The  first  day 
A  laid  2,136  bricks,  and  the  second  day  he  laid  as  many 
as  the  first  day  plus  207.  B,  on  the  first  day,  laid  1,936, 
anTl,  on  the  second  day,  341  more  than  on  the  first.  How 
many  bricks  were  laid  by  both  in  the  two  days? 

24.  The  distance  from  Greening  to  Chatfield  is  277 
miles,  from  Chatfield  to  Glendale  is  325  miles,  from  Glen- 
dale  to  Wyoming  is  139  miles,  from  Wyoming  to  Dale  is 
193  miles.  By  this  route  what  is  the  distance  from  Green- 
ing to  Dale? 

25.  A  man  took  2,126  steps  going  from  home  to  his 
place  of  business,  3,197  while  in  his  store,  6,239  going 
fn)m  there  to  the  jmrk,  5,782  while  in  the  park,  8,573 
going  from  there  home.  What  was  the  whole  numljer 
of  steps  taken  by  him  from  the  tinio  lu*  left  until  he  re- 
entered his  house? 


30 


ADDITION. 


26.  In  the  first  story  of  a  house,  the  hall  contained  117 
square  feet,  the  parlor  327,  the  sitting-room  296,  the  dining- 
room  257.  How  many  square  feet  of  carpeting  would  be 
rcHjuired  to  cover  the  lloors  of  these  rooms? 

27.  In  1870,  the  population  of  Buffalo  was  117,714;  that 
of  Rochester,  62,386;  that  of  Albany,  69,422;  that  of  Brook- 
lyn, 396,099.  How  many  inha])itnnts  did  these  four  cities 
contain? 

28.  The  area  of  Spain  is  r.».>,7<.>  s<|uare  miles;  that  of 
France,  204,091 ;  that  of  Switzerland,  15,922;  that  of  Italy, 
112,622.  Over  how  many  8(]uare  miles  do  the  four  countries 
extend? 

29.  A  speculator  bougl.  ;  »t8  for  $1,375  each.  He 
sold  the  first  for  $125  more  than  cost,  the  second  for  $319 
more  than  cost,  the  third  for  $291  more  than  cost,  the  fourth 
for  $739  more  than  cost,  and  the  fifth  for  $135  more  than 
cost     How  much  money  did  he  receive  for  all  ? 

30.  The  State  of  Alabama  contains  1,430  libraries  and 
576,882  volumes;  Mississippi,  2,788  libraries  and  488,482 
volumes;  Louisiaim,  2,332  libraries  and  847,406  volumes; 
Texas,  455  libraries  and  87,111  volumes.  How  many  libra- 
ries and  how  many  volumes  do  the  four  States  contain? 

31.  The  population  of  five  of  the  principal  cities  of  Ohio 
was  in  1870  as  follows:  Cincinnati,  216,239;  Cleveland, 
92,829;  Toledo,  31,584;  Columbus,  31,274;  Dayton,  30,473. 
What  was  the  entire  population  of  these  cities  in  1870? 

32.  The  population  of  five  of  the  principal  cities  of  Illinois 
was  in  1870  as  follows:  Chicago,  298,977;  Quincy,  24,025; 
Peoria,  22,849;  Springfield,  17,364;  Bloomington,  14,590. 
What  was  the  entire  population  of  these  cities  at  that  time? 

33.  In  1870,  the  population  of  St.  Louis,  Mo.,  was  310,864; 
Memphis,  Tenn.,  40,226;  Charleston,  S.  C,  48,956;  Rich- 
mond, Va.,  51,038;  New  Orleans,  La.,  191,418.  What 
was  the  entire  population  of  these  cities  at  that  time? 


ADDITION.  ;jl 

84.  The  Warsaw  Manufacturing  Company  nawed  11,936 
feet  of  pine  on  Monday,  12,117  feet  of  hemlock  on  Tues- 
day, 8,135  feet  of  maple  on  Wednesday,  and  9,963  feet  of 
ash  on  Thursday.  How  many  feet  of  timber  did  they  saw 
in  the  four  days? 

35.  According  to  the  census  of  1870,  the  number  of  native 
Americans  in  Nebraska  was  92,245;  the  number  of  Irish, 
4,999;  of  Germans,  10,954;  of  English,  3,602;  of  Scotch, 
792;  of  Canadians,  2,632;  of  French,  340;  of  Norwegians, 
506;  of  Swedes,  2,352.  What  was  the  total  population  of  the 
State  in  1870? 

36.  In  a  certain  State  there  were  raised,  last  year, 
7,771,009  bushels  of  potatoes,  278,798  bushels  of  wheat, 
1,089,888  bushels  of  Indian  corn,  2,351,354  bushels  of 
oats,  658,816  bushels  of  barley.  What  was  the  entire 
number  of  bushels  of  farm  products  raised  that  year? 

37.  Mr.  George  Peabody  gave  to  the  poor  of  London 
^2,250,000,  to  the  town  of  Danvers  $60,000,  to  the  Grin- 
nell  Arctic  Expedition  $10,000,  to  the  city  of  Baltimore 
$1,000,000,  to  Phillips'  Academy  $25,000,  to  the  IMassa- 
chusetts  Historical  Society  $20,000,  to  Harvard  College 
$1-50,000,  to  Yale  College  $150,000,  to  the  Southwest 
$1,500,000.  How  much  did  this  benevolent  gentleman 
give  away? 

38.  In  1870,  there  were,  in  the  United  States,  574  daily 
newspapers,  with  a  circulation  of  2,601,547;  107  tri-weck- 
lies,  with  a  circulation  of  155,105;  115  semi-weeklies,  with 
a  circulation  of  247,197;  4295  weeklies,  with  a  circulation 
of  10,594,643;  96  semi-monthlies,  with  a  circulation  of 
1,349,820;  622  monthlies,  with  a  circulation  of  5,650,843; 
13  bi-monthlies,  with  a  circulation  of  31,650;  49  quarter- 
lies, with  a  circulation  of  211,670.  How  many  perimlicals 
were  there  in  the  United  States  during  that  year,  and  what 
was  their  entire  circulation? 


32  ADDITION. 

39.  Mr.  A.  deposited  iu  the  First  National  Baiik'of  Albany, 
N.  Y.,  on  July  3,  1877,  $395.25;  on  July  5,  8874.75;  on 
July  8,  $325.85.  He  also  deposited  in  the  National  Park 
Bank  of  New  York  City,  on  July  12,  1877,  $1,540.87;  on 
July  16,  $1,275;  on  July  20,  $1,985.50.  How  much  did  he 
deposit  in  each  of  tlio  lianks?    How  much  in  both  banks? 


(40.) 

(41.) 

(42.) 

(43.) 

2134 

6166 

5873 

46321 

8060 

5878 

3858 

69788 

5032 

9H76 

6430 

76434 

8797 

7977 

5082 

68924 

9888 

0503 

6353 

96355 

6432 

4556 

4202 

88789 

5421 

6432 

8792 

93745 

(44.) 

(45.) 

(46.) 

(47.) 

813 

7(50 

3945 

5063 

976 

500 

9204 

2050 

432 

750 

8769 

3254 

397 

694 

9876 

4200 

788 

942 

8020 

6131 

643 

293 

5612 

5945 

664 

978 

3424 

2763 

321 

785 

5861 

4828 

156 

696 

2188 

7688 

642 

785 

7654 

3288 

321 

688 

3210 

5634 

876 

762 

8765 

6546 

543 

451 

5849 

3250 

429 

984 

8574 

7864 

386 

579 

9836 

9758 

595 

384 

8759 

8410 

(P\i 


Ss^i;^^ 


SUBTRACTION 


h 


I  I    I      I    \  I   i: 


55.    1.  If  I  liuve  ()  i)eaches  and  give  away  3  of  them,  how 
manv  will  be  left? 

2.  If  James  has  4  bunches  of  grapes  and  eats  2  of  tliem, 
how  many  will  be  left? 

3.  If  I  have  7  bunches  of  grapes  and  give  away  4  ol  ' 
them,  how  many  will  Ixi  left? 

4.  If  you  find  8  acorns  and  lose  4  of  them,  how  many 
will  be  left? 

o.  How  many  are  left  when  4  things  are  taken  from  8 
things?  How  many  are  8  less '4?  7  Ic^s  \'i  ."i  ]( ><  ^y  !> 
Ies8  4?     G  less  4? 

6.  A  farmer  who  had  7  horses,  sold  3  of  them.  How 
many  had  he  left?     How  many  are  7  less  3?     9  less  3? 

7.  James  earned,  during  the  summer,  $9.  He  spent  85 
of  the  money  for  a  coat,  and  the  rest  for  a  pair  of  boots. 
How  much  did  the  boots  cost  him? 

8.  Nine  is  how  many  more  than  5?  Than  6?  Than  4? 
Than  3? 

9.  A  buy  who  had  'J  cliicki  ii.-,  .sold  o  (»!'  tluiii.  Ilnw 
many  had  he  left? 

10.  Lawrence  had  10  pictures  in  his  room.     He  gave  his 
sister  3  of  them.     How  many  were  left  in  his  room? 

11.  A  man  earned  $11   per  week  and  spent  $7.     How 
much  did  lie  >ave  weekly? 

■^  (33) 


34  SUBTRACTION. 

1:.'.  A  hen  had  nine  chickens,  but  5  of  tliem  were  killed. 
How  many  chickens  were  left?  How  manv  mu-t  ixi  added 
to  5  to  make  9? 

13.  When  5  is  taken  IV  hat  numlxjr  is  left? 

14.  When  7  is  taken  Irom  10,  what  number  remains? 
How  many  must  be  added  to  7  to  equal  10? 

15.  Howard  is  10  years  of  age  and  Horliort  .  Wiiat 
is  the  dirt'ercncc  in  their  ages?  What  i>  th.  ditierence  be- 
tween 10  and  8? 

16.  If  the  difference  betw.  lod  to  8, 
what  will  the  result  be? 

17.  If  the  difference  between  ui..     ....  ...........  U  ;ukkd 

to  the  smaller  number,  to  what  will  the  result  !«  equal? 

*    18.  What  is  the  difference  between  G  horses  and  4  horses? 
Between  6  horses  and  5  cents? 

19.  Why  can  you  not  find  the  difference  between  0  horses 
and  5  a 

20.  \\  -  .limnmce  lx.i  --1 
4  horse- 

21.  lietweea  what  kin  mbers  only  <  an  ilie  dilicr- 
ence  be  found? 

56.  Subtraction  r  lakinir  "n.-  mimlier 
fn)m  another. 

57.  The  Minuend  is  the  numb<  r  !il(  li  aimtlRr 

subtracted. 

."js.    I  he  Subtrfthend  is  the  number  to  be  subtracted. 

59.  The  Reuui hitler,  i  J^iffereuv*\  \>  the  result 
obtained  by  subtracting. 

00.  The  Sign  of  Subtraction  is  a  short  horizontal 
line:  — .     It  is  named  mimvi. 


SUBTRACTION. 


35 


When  the  sign  minus  is  placed  between  two  nunil)crs  it 
shows  that  the  one  after  it  is  to  be  subtracted  from  the  ono 
before  it 

Thus,  ivad  9  luinuti  5,  and  means  that  5  is  to  be  sub- 

tracted from  y. 

()1.  PiuxciPLE. — 1.  Only  like  numbers  can  he  mbtracted.    • 
2.   Tlie  sum  of  Vie  subtrahend  ami  remainder  mud  be  equal  to 
the  minuend. 

TABLE. 


1-1=  0 

2-2-  Oi  3  3-0 

4—4=  0  5—  5=  0 

2—1=  1 

3-2=  1 

4-3=  1 

5—4=  1  6—  5=  1 

3—1=  2 

4—2=  2 

5-3=  2 

6—4=  2  7—  5=  2 

4-1=  3 

6—2=  3 

6-3=  3 

7-4=  .  -   5=  3 

5-1=  4 

6—2=  4 

7—3=  4 

8-4=  4  y—  5=  4 

G-l=  5 

7-2=  5 

8-3=  5 

9—4=  5  10—  5=  5 

7—1=  6 

8-2=  6 

9—3=  6 

10—4=  6|11— 5=  6 

8—1=  7 

9—2=  7 

10-3=  7 

11—4=  7 

12-  5=  7 

9—1=  8 

10-2=  8 

11-3=  8 

12—4=  8  13—  5=1?  8 

10-1=  9 

11—2=  9112-3=  9 

13-4=  9  14-  5=  9 

.11—1=10'  12-2=10  13-3-:10 

14-4^10 

15—  5=10 

6-6=  0 

7—7=  0:  8-8=  0 

"9  *.'   •' 

10—10=  0 

'  7-6-  1 

8-7=  1 

9-8-^  1 

10—9=  1 

11-10=  1 

8— (r=  2 

9-7=  2 

10-8=  2 

11-9=  2 

12—10=  2 

9-6=  3 

10-7=  3 

11—8=  3 

12-9=  3 

13-10=  3 

10-6=  4 

11-7=  4 

12-8=  4 

13-9=  4 

14—10=  4 

11-6 -.  0 

12-7=  5 

13-8=  5 

14-9=  5 

15-10-^  5 

12  6   <i 

l.T  7  -  6 

14-8=  6 

15-9=  6  16— 10=  6 

13—6=.  7 

14-7=  7 

15-8=  7 

16-9=  7 1 17-10=  7 

14-6=  8 

15-7=  8 

16-8=  8 

17-9=  8)18-10=  8 

15—6=  9 

16-7=  9 

17_8.-^  9 

18-9=  9 1 19-10=  9 

16-6-10'l7  7-  lO'lM  «  10 

lu  o-lO  20  -  10  10 

36  SUBTRACTION. 


rA?;E    T. 

0*2.  WIk'ii  no  fiKiirc  of  Hk' Miibtrulicud  liaM  a  ii^roater 
TaliK'  lliaii  tii<'  i*orr4>K|><»ii<liii};  fiu;iire  of*  the  iiiiiiiK'iid. 

1.  A    :     reliant  had  15  barrels  of  flour,  and  sold  4  of 
tftem.     How  many  had  he  led? 

Analysis — Since  he  had  15  barrels  of  flour  and  sold  4  ot  them,  he 
had  left  the  difTerence  between  15  barrels  and  4  hurix>l8,  which  is  11 
barrels. 

2.  Ali«  «•  lM)UL'ht  18  cakes,  and  ate  6  of  them.    How  many 
had  she  hfi? 

3.  Jame^  saw  17  hirils  on  a  tree,  but  7  soon  flew  away. 
How  many  remuincii? 

4.  If  a  man  I'arns  S19  a  week,  and  spends  $9,  how  much 
wiU  he  save  each  week? 

5.  Lewis  owed  his  brother  $7,  and  paid  him  Srj.     How 
much  did  he  still  owe  him? 

6.  Eliza  had  16  plums,  but  gave  5  to  her  father.     How 
many  had  she  left? 

7.  James  had  $12,  and  lost  $2.     How   many  had  he 
left? 

8.  If  John  is  19  years  old,  and  Ma«r*ri'    !"•.  li'W  much 
younger  than  John  is  Maggie? 

9.  In  the  same  shop  6  boys  and  17  men  work.     How 
many  more  men  than  boys  are  there  in  the  shop? 

10.  There  were  18  girls  and  7  boy>  in  a  clas,*.  How 
many  more  girls  than  boys  were  then 

11.  Laura  had  14  cents,  and  lost  3  cents.  How  many 
had  she  then? 

12.  Henry  solved  19  examples,  and  George  solved  8. 
How  many  more  did  Henry  solve  than  George? 

13.  William  wrote  16  lines  in  his  copy-book,  and  Peter 
wrote  5  lines  less.     How  many  did  Peter  write? 


SUBTRACTION.  37 

14.  One  piece  of  cloth  contained  20  yards  and  another  10 
yards.     How  many  yards  more  were  tliere  in  the  larger  piece? 

15.  A  boy  had  24  chickens,  and  10  of  them  died.     How 
many  had  he  left? 

.10.  Juliii  gave  me  11  cents.     If  she  had  16  cents  at  first, 
liow  many  had  she  left? 

17.  A  girl  bought  18  eggs,  and,  on  her  way  homo,  foil  and 
broke  5  of  them.     How  many  had  she  left? 

18.  Subtract  by  2*8  from  22  to  0;   thus:  22,  20,  18,  16, 
14,  12,  etc. 

19.  Subtract  by  3*8  from  35  to  2.     From  45  to  0. 

20.  Subtract  by  4's  from  48  to  0.     From  45  to  1. 

21.  Count  back  by  5's  from  35  to  0.     From  59  to  4. 

WRITTEN    EXEnCISES, 

63.   1.  From  547  subtract  235. 

PROCESS.  Analysis. — For  convenience  we  write  the  less 

Minuend       5  4  7      mmiber  under  the  greater,  units  under  units,  tens 

r,  ^.    X.     J   o  o  -       under  tens,  etc.,  and  subtract  each  order  of  units 
Subtrahend   260  ,     ,  ,  ,        ,    .         . 

separately  from  tlie  same  order  of  the  minuend. 

Remainder    312  Thus,   7    units  —  5   units  =  2  units,  which  we 

write  under  the  units. 

4  tens  —  3  tens  =  1  ten,  which  we  write  under  the  tens. 

5  liundretls  —  2  liundreds  ::=  3  hundreds,  which  we  write  under  the 
hundrc<ls. 

Hence  tiie  rtiuuintkr  is  ol2. 

Proof. — 312,  the  remainder,  plus  235,  the  subtrahend,  equals  547, 
the  minuend.    Hence  the  result  is  correct. 

Copy,  subtract,  and  prove : 


(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

(7.) 

713 

458 

986 

854 

795 

7842 

302 

134 

732 

641 

433 

2310 

38 


sifKrr'  \i  1  I, 

>  \  . 

(8.) 

(9. 

M'.; 

,11.) 

(12.) 

$48.25 

$64.  L' 

!").78 

$38.94 

$41.89 

23.13 

30.  J 

;i.65 

27.83 

20.45 

13.  A  drover,  having  -heep,  soM  \  \M  nl  tlinn 
How  many  had  he  left? 

14.  A  speculator  bought  some  land  for  $5841.1.75,  and  sold 
it  for  $0959.95.     How  much  did  he  gain? 

I  '.  A  cotton  fiictory  made  9875  yards  of  cloth  in  one 
week,  and  aold,  during  the  8am<"  iiin< .  7652  yards.  How 
much  more  was  made  than  sold? 

16.  A  money-lender  received  for  interest,  during  1875, 
$1685.49.  and  during  1876,  $2796.59.  In  which  }Hjar  did 
ho  HMvivo  the  greater  sum,  and  how  much? 

17  A  iii:in  bought  7467  bricks,  and  carted  away  3136. 
ilow  many  remained  to  be  moved? 

18.  A  h:is  736  sheep,  and  B  hiif*  213  le??  than  A.  How 
many  sheep  has  B? 

19.  A  man  gave  his  note  for  .-u7:»j,  wimout  inunst. 
In  two  years  he  had  paid  $3401.  How  much  did  he  .still 
owe  on  the  note? 

20.  A  man  bought  a  house  for  >i7t).),  and  afterward  .sold 
it,  thereby  losing  $504.     For  how  much  did  he  sell  it? 

21.  A  man  bought  a  span  of  horses  for  $364,  and  a  yoke 
of  oxen  for  $120.  How  much  mnro  did  ho  L'ivo  ff)r  the 
horses  than  for  the  oxen? 

22.  A  merchant  having  6755  yards  of  cloth,  sold  2532 
yards.     How  many  yards  had  he  remaining? 

23.  A  father  having  3052  acres  of  land,  gave  his  son  1230 
acres.     How  many  acres  had  he  left? 

24.  A  vintner  had  38756  gallons  of  wine,  and  sold  during 
the  year,  34243  gallons.     How  much  remained  unsold? 

25.  The  circulation  of  a  newspaper  in  1875  was  38293 
copies,  and  in  1870,  37180.     What  was  the  decrease? 


SUBTRACTION.  39 


CASE  II. 

64.  H'liPU  au^  figure  of*  the  subtrahend  has  a  greater 
value  tlBau  tlie  e4»rreN|>ou<liiig  figure^  of  the  minuend.    . 

1.  A  geutleniau  bought  a  coat  at  §40,  and  a  vest  at  §9; 
he  gave  the  merchant  a  hundred-dollar  bill.  How  vwich 
change  ouglit  he  to  receive? 

Analysis. — Since  he  paid  S40  for  a  coat  and  $9  for  a  vest,  for  both 
he  paid  the  smn  of  $40  and  $*J,  or  $49.  And  since  he  jjave  the  luer- 
cltant  $100,  he  Khould  receive  the  difference  between  $100  and  $49. 
t^io.)— ^viu  — ;.SGO;  $00— $9r=$51.    Therefore  he  should  receive  $51. 

2.  A  boy  saw  15  birds  on  a  tree,  and  9  of  them  flew  away. 
How  many  remained? 

3.  John  is  16  years  old,  mid  .Innn  >  i-  >.  II  'W  mucli  elder 
than  James  is  John? 

4.  A  jeweler  lx)ught  a  watch  for  875,  and  sold  it  lor  $100. 
How  much  did  he  gain  by  the  operation? 

A  grocer  bought  a  quantity  of  sugar  for  836,  and  re- 
tail cl  the  same  for  $50.     How  much  did  he  gain  by  the  siile? 

6.  A  boy  had  34  marbles,  and  gave  away  9  of  them. 
How  many  had  he  left? 

7.  A  lady  bought  a  chair  lor  $»'),  and  a  tabic  ior  sf5;  she 
gave  a  twenty-dollar  bill  to  the  cabinet-maker.  How  much 
change  ought  she  to  receive? 

8.  A  man  set  out  to  walk  50  miles;  he  walked  20  miles 
the  first  day,  and  19  the  second  day.  How  many  miles  were 
lefl  for  him  to  walk? 

9.  A  man  iMuight  a  cow  for  $35,  and. sold  her  for  $43, 
af\er  keepinir  l>er  4  week-  i  :iii  expense  of  $2  per  w..  k. 
How  much  AU\  he  giiin? 

10.  A  man  who  earned  $60  a  month,  paid  $25  a  month  for 
hi.<«  IxMird,  and  Slo  a  month  for  other  expenses.  How  nuirli 
did  he  sjwc  |)cr  month? 


10  SUBTRACTION. 

11.  Count  back  by  10*8  from  107  to  7;  thus:  lu7,  'J7,  etc. 

12.  Count  back  by  10*8  from  95  to  5.     From  79  to  9. 
18.  Count  back  by  lO's  from  83  to  13.     From  98  to  18. 

1-1.  SuhtniL-t  by  2(rs  from   lOU  to  20:   thus:  IOC),  86,  ete. 

WJC  I   /    /   /     ^  /    s . 

65.    1.  From  643  subtract  456. 

rR(x:Es».  Analysi.h. — Wc  write  the  numbeni  an  in  the  previous 

Q ^  >}  ta>*c  and   begin  nt  the  right  to  subtract. 

.  -  Since  6  uniUi  can  not  be  subtracted  from  3  units,  we 

iiitite  with  the  3  units  a  unit  of  the  next  higher  order, 

187  which  is  equal  to  10  units,  making  13  units:  6  units  from 

13  units  leave  7  units,  which  we  write  under  the  units. 
Since  one  of  ttns  was  united  with  tlie  unitM,  there  can  l)e  but  3  tens 
tt.     Because  6  ten«  can  not  be  subtracted  from  3  tens,  we  unite  with 
iu>  3  tens  as  befon>,  a  unit  of  the  next  higher  order,  which  is  khkiI 
lit   10   tens,  making  13  tens:  6  tens    from    13   tens   leave 
winch  we  write  under  the  tena. 

Since  one  of  the  hundreds  was  united  with  the  tens,  there  are  but  5 
hundreds  left:  4  hundre<Is  from  5  hundreds  leave  1  hundred,  which 
wo  write  under  the  hundreds.    Henoe  the  result  is  187. 

Proof. — 187,  the  remainder,  plus  456,  the  subtrahend,  equals  643, 
the  minuend.    Hence  the  result  is  correct. 

60.  Rule. — Write  the  subtrahend  under  fhr  minvnifl  — '^^ 
uitdrr  uuiti*,  tens  under  tens,  etc 

Be(jin  at  the  right  and  tvbtraet  each  figttre  of  the  subtrahend 
from  the  corresponding  figure  of  th*"  *"''«"*'??^?,  tn-ifhifj  the  result 
beneath. 

If  «  /srwc  in  the  minuend  hn<  n  A>>  value  tJian  the  corre- 
ffjyonding  figure  in  the  subtrahend,  iticna^^  the  farmer  by  ten,  and 
subtract;  then  diminish  by  one,  the  units  oftlie  next  higher  order  in 
the  minuend,  and  subtract  as  before. 

Proof. — Add  together  the  remainder  and  subtraJiend.  If  the 
residt  be  equal  to  tlie  minuend  the  work  is  correct. 


SUBTRACTION. 


41 


EXAMPLES. 

Copy,  subtract,  and  prove: 

r-'.) 

(3.) 

(4.) 

(5.) 

(6.) 

75."> 

1)84 

826 

754 

1426 

44.S 

756 

534 

482 

547 

(7.) 

(8.) 

(9.) 

(10.) 

(11.) 

843 

1846 

1683 

2897 

3001 

782 

927 

1395 

1598 

2851 

(12.) 

(13.) 

(14.) 

(15.) 

(16.) 

$24.45   $39.18 

863.25 

$71.89 

$42.34 

21.3^ 

i    27.92 
3  difference  betv 

47.18 

.  47.93 

18.67 

Find  th( 

een 

17. 

583  and  294. 

25. 

7812  and  1984. 

18. 

690  and  508. 

26. 

8003  and  5872. 

-   19. 

763  and  574. 

27. 

63004  and  54872 

20. 

966  and  599. 

28. 

65432  and  54862. 

21. 

982  and  796. 

29. 

69721 

and  49653. 

22. 

891  and  798. 

30. 

78303  and  49424. 

23. 

5833  and  4968. 

31. 

865932  and  785841, 

24. 

7521  and  3635. 

32. 

9050308  and  563420. 

33.  A  man  set  out  on  a  journey  of  861  miles.  During  the 
first  day  he  traveleil  297  miles,  and  during  the  second  day 
308  miles.     How  many  miles  had  he  yet  to  travel? 

34.  A  merchant  de|)osited  in  a  bank  on  Monday  $584;  on 
Tuesday,  8759;  on  Wednesday,  $463.  He  drew  out  $1298 
during  that  time.  How  much  did  his  deposits  'exceed  what 
he  tlrow  out? 


42 

35.  A  grocer  had  871.3  ^xjuiids  ut  sugar  on  Imnd.  On  one 
day  he  sold  1235  poundi*,  on  the  next  1317;  the  third  day 
he  sold  to  C  all  the  sugar  that  remained.  How  manv  pounds 
did  C  buy? 

36.  I  bin^ht  a  horse  for  $637,  and  a  cow  for  S317.  I  sold 
the  hor  7  JO,  and  the  -  ^  How  much  did 
I  gain  l)y  iik-  saie? 

37.  In  the  first  of  three  pavements  there  are  1317  bricks, 
ui  the  second  there  are  2357,  in  the  third  there  are  1719 
less  than  in  both  *■  '  V.  my  bricks  in  the  third 
pavement? 

38.  In  1869  there  were  264,146,yuu  bushels  of  wheat  raised 
in  the  United  States,  and  874,120,005  bushels  of  <  urn.  How 
much  more  corn  than  wheat  wa.s  produced? 

V  l)ought  351  acres  of  land,  and  B  bought  27  acres 
nioiv  man  A;  B  sold  his  land  to  C,  who  tl:  '  '  "T  Mcres. 
How  many  acres  did  C  have  at  first? 

40.  A  grocer  retailed  a  quantity  of  sugar  lor  ^308.40,  and 
80  gained  8106.28.     How  much  had  he  paid  for  it? 

41.  The  year  1870  was  just  378  years  after  the  discovery 
of  Americui  by  Columbus.  In  what  year  did  that  event  take 
place? 

42.  On  Monday  morning  a  bank  had  on  hand  $1826.  Dur- 
ing the  day  S2191  were  deposited  and  $3412  drawn  out;  on 
Tuesday  $3256  were  deposited  and  $2164  drawn  out.  How 
many  dollars  were  on  hand  Wednesday  morning? 

43.  R.  S.  Hill  is  worth  $15795,  of  which  $2895  is  in- 
vested in  bank  stock,  $3864  in  mortgiiges  and  the  re«t  in 
land.     How  much  has  he  invested  in  land? 

44.  Of  the  two  numbers  89346  and  56849,  how  much 
nearer  is  the  one  than  the  other  to  68754? 

45.  The  numlx^r  of  pupils  who  attended  school  in  Boston 
in  1870  was  38944,  and  of  this  number  35442  attended  the 
public  scho'>]<.     How  many  attended  the  other  «ohMo]s? 


r  MULTIPLICATION 


INDUCTirX    EXERCISES, 

67.  1.  How  many  books  are  there  in  2  piles  containing  3 
books  each? 

2.  If  you  place  4  apples  in  a  group,  how  many  apples  are 
there  in  o  such  groups?     In  4  groups? 

3.  When  there  arc  3  roses  in  a  cluster,  how  many  ai« 
there  in  3  clusters?     In  4  clusters?    In  5  clusters? 

4.  How  many  are  3  +  3-f  3-{-3,  or  four  3's? 
6.  How  many  are  4  +  4-|-4,  or  three  4's? 

6.  How  many  are  four  4's?     Four-5's?     Four  6's? 

7.  James  bought  5  pencils  at  5  cents  each.  How  much 
did  they  cost  him?  How  many  cents  are  5  times  5  cents? 
How  many  are  five  5's? 

8.  An  orchard  contains  5  rows  of  6  trees  each.  How 
many  trees  are  there  in  the  orchard?  How  many  trees  are 
5  times  6  trees?     How  many  are  5  times  6? 

9.  James  piled  his  blocks  in  3  piles,  each  containing  5 
blocks.  How  many  blocks  had  he?  How  many  are  3  times 
5  blocks?     How  many  are  3  times  5? 

10.  A  boy  earned  84  a  week  for  6  weeks.  How  much 
did  he  earn  in  all?  How  many  dollars  are  6  times  84? 
How  many  are  6  times  4? 

11.  Harry  played  5  hours  per  day.  How  many  hours  did 
he  play  in  G  days?  How  many  are  6  times  5  hours?  How 
many  are  6  times  5? 

(43) 


44  MULTIPUCATION. 

12.  How  does  5  times  4  compare  with  4  times  5?    5  times 
6  with  6  times  5? 

13.  When  numbers  are  u»cii  wiiliont  nlrronee  to  anv  par- 
ticuhir  thing,  they  are  called  Abatract  ^umbem. 


DEFINITIONS. 

(iH.  Multiplicatimt  is  a  short  process  of  finding  the 
sum  of  equal  numben^ ; 

The  process  of  repealing  <»ue  number  as  many  times  as 
there  are  units  in  another. 

69.  The  Multiplicand  is  the  number  to  be  repeated 
or  multiplied. 

70.  The  Multiplier  is  the  number  showing  how  many 
limes  the  multiplicand  is  to  be  repeated. 

71.  The  Product  is  the  result  obtained  by  multiplying. 

72.  The  multiplicand  and  multiplier  are  called  the  faelan 
of  the  product, 

7H.  The   Siij^n  of  Multiplication  is   an   oblique 
It  is  read,  multiplied  by,  or  times.     When  placed 
'.  iMti  ii  iwo  numbers  it  shows  that  they  are  to  be  multiplied 
t(ig<*ther. 

rim  .  t  X3  is  read,  4  maltiplied  by  3,  or  3  times  4. 

74.  Principles. — 1.  The  mtilttplier  mtist  he  regarded  as  an 
abstract  number. 

2.  The  multiplicand  and  prodtut  must  be  like  numbers. 

3.  Either  of  tiie  factors  may  he  used  as  multiplicand  or  multi- 
plier when  botii  are  abstrad. 

In  practice,  for  convenience,  the  smaller  number  is  generally  used 
as  multiplier. 


MULTIPLICATION. 


45 


TABLE. 


ix  1=  1 

2X  1=  2 

3X  1==  3 

4X  1=  4 

5X  1=    5 

IX  2=  2 

2X  2=4 

3X  2=  6 

4X  2=  8 

5X  2=  10 

IX  3=  3 

2X  3=  6 

3X  3=  9 

4/  3=12 

5X  3=:t  15 

IX  4=  4 

2X  4=  8 

3X  4=12 

4X  4=16 

5X  4=  20 

IX  5=  5 

2X  5=10 

3X  5=15 

4X  5=20 

5X  6=  25 

IX  6=  6 

2X  6=12 

3X  6=18 

4X  0=24 

5X  6=  30 

IX  7=  7 

2X  7=14 

3X  7=21 

4X  7=28 

5X  7=  35 

IX  8=  8 

2X  8=16 

3X  8=24 

4X  8=32 

5X  8=  40 

IX  9=  9 

2X  9=18 

3X  9=27 

4X  9=30 

5X  9=  45 

1X10=10 

2x10=20 

3x10=30 

4X10=40 

5x10=  50 

6X  1=  6 

7X  1=  7 

8X  1=  8 

9X  1=  9 

lOx  1=  10 

6X  2=12 

7x  2=14 

8X  2=16 

9X  2=18 

lOX  2=  20 

6X  3^18 

7X  3=21 

8X  3=24 

9X  3=27 

lOx  3=  30 

6X  4=24 

7X  4=28 

8X  4=32 

9X  4=36 

lOx  4=  40 

6X  5=30 

7X  5=35 

8X  5=40 

9X  5=45 

lOx  5=  50 

6X  6=36 

7X  6=42 

8X  6=48 

9X  6=54 

lOx  6=  60 

6X  7=42 

7X  7=49 

8X  7=56 

9x  7=63 

lOx  7=  70 

6X  8=48 

7X  8=56 

8x  8=64 

9X  8=72 

lOX  8=  80 

^X  9=54 

7X  9=63 

8X  9=72 

9X  9=81 

lOx  9=  90 

6X10=60 

7X10=70 

8X10=80 

9X10=90 

10x10=100 

CASE  I. 
75.    lillioii      tlio    iiiii]ti|»]i<'r    in     vxpn^ssetl     by     ouc 

1.  What  will  5  yards  of  ribbon  cost  at  6  cents  a  yard? 

Analysis. — Since  1  yard  of  ribbon  costs  C  cents,  5  yards  will  coat 
5  timeA  6  centa,  or  30  cents. 

2.  At  7  cents  each,  what  will  8  pencils  cost? 

3.  What  will  be  the  cost  of  8  melons  at  9  cents  each? 


46  MULTIPLICATION. 

4.  ^Vhat  will  V}  sheep  cost  at  $6  a  head? 

5.  When  p:m|>eiJ  are  worth  7  cents  a  bunch,  h<t\\  nuifh 
will  9  bunches  c<j.«t? 

6.  James  bought  5  nibbita  for  7  dimes  each,  and  sold 
them  for  5  dimes  each  more  than  he  jmid  for  them.  How 
much  did  he  gain?     How  much  did  he  get  for  them? 

7.  How  much  did  he  get  for  each  rabbit?  How  does  5 
times  12  com|)are  with  5  times  7  pliu  5  times  5? 

8.  How  many  are  o  times  15,  or  5  times  10  plm  5  times  5? 

9.  How  many  are  3  times  18,  or  8  times  10  plus  3  times  8? 

10.  James  sold  7  dove?  *'  ~  "* '        M       nmch  did 

he  get  for  them? 

11.  What  will  be  the  cost  of  »  slates  at  15  cent-  ? 

12.  If  one  inkstand  costs  18  cents,  how  much  will  4  i^jsii' 

13.  A  lady  bought  7  yards  of  riblwn  at  19  cent-'  n  yard. 
How  much  did  she  pay  for  it'/ 

14.  If  a  man  walked  IH  hm;  -  m  ■  m  min.  n  a  in  (onld 
he  walk  in  3  days? 

15.  How  much  can  a  boy  earn  in  8  hours  if  he  earns  13 
cents  in  an  hour? 

16.  How  many  arc  5  times  20?  5  times  7?  5  times  20 
plus  5  times  7,  or  5  times  27? 

17.  How  many  are  5  times  23,  or  -f  Liim  -  2^'  jn,..^  .,  i.i.ics  3? 

18.  How  many  are  7  times  24,  or  7  times  20  plus  7  times  4? 

19.  How  many  are  9  times  22,  or  9  times  20  plus  9  times  2? 

20.  If  there  are  25  yards  in  f»nc  ])ioco  of  <'l<>th,  how  many 
yards  are  there  in  8  pieces? 

21.  A  boy  bought  7  chickens  at  27  How 
much  did  he  j)ay  for  all? 

22.  If  a  steamship  can  sail  28  miles  in  one  hour,  how  far 
can  she  sail  in  8  hours? 

23.  If  the  cars  run  26  miles  an  hour,  how  liir  will  they 
run  in  5  hours?  How  far  in  6  hours?  How  far  in  7 
hours? 


MULTIPLICATION.  47 

24.  If  6  men  can  do  a  piece  of  work  in  21  days,  how 
long  will  it  take  one  man  to  do  the  same  work? 

25.  In  a  certain  orchard  there  are  9  rows  of  trees  and  27 
trees  in  a  row.     How  many  trees  are  there  in  the  orchard? 

20.  ('ount  by  2'8  from  0  to  24;   thus:  2,  4,  C,  8,  10,  etc. 

27.  Count  by  3's  from  0  to  36.     By  4's  from  0  to  48. 

28.  Repeat  all  the  numbers  of  times  5  from  once  5  to  10 
times  5.  Thus,  once  5  i<  .",  :.>  times  5  are  10,  3  times  5  are 
15,  etc. 

29.  Repeat  from  oiice  (i  to  10  times  6,  and  back  from  10 
times  6  to  once  6. 

30.  Re]x?at  from  once    7  to  10  times    7,  and  reverse. 

31.  Repeat  from  once    8  to  10  times    8,  and  reverse. 

32.  Re}xjat  from  once    9  to  10  times    9,  and  reverse. 

33.  Repeat  from  once  10  to  10  times  10,  and  reverse. 

34.  At  25  cents  a  pound,  how  much  will  6  pounds  of 
raisins  cost? 

35.  If  a  man  can  dig  28  bushels  of  potatoes  in  one  day, 
how  many  can  he  dig  in  4  days? 

36.  If  a  i^erson  spend  25  cents  a  day  for  cigars,  how  much 
will  he  si)end  in  7  days? 

37.  If  a  lH)y  earns  33  cents  a  day,  how  nuidi  will  he  earn 
in  9  days? 

38.  When  butter  is  selling  at  37  cents  a  jwund,  wliat  will 
7  pour.ds  cost  me? 

ir  JtJTT E \   E X  1-: mis  i:  s . 

7G,  1.  lluw  nituiy  are  3  times  275? 

Analysis. — Since  multiplication  is  a  short 
jirocess  of  adding  c(iual  nuinber8,  it  is  evident 
that  we  can  determine  by  addition  how  many  3 
times  275,  or  three  275's,  are.    The  sum  is  825. 

In  urnctice,  a  shorter  method  is  employed, 
which  is  given  in  the  second  process  and  analysis. 


1st 

PROCES.' 

275 

275 

275 

Swn 

825 

48 


MULTfPIJCATION. 


i>  PBOCESB. 


Mult 
Mull 


Analysw. — For  convenience   we   write    the 

- -:1tiplier  under  the    nuiliipUeand,  and   begin 

the  right  to  multiply.    Thus,  'A  times  5  unit« 

art-   15   unitM,  or  1    ten   and   5  unitn.     We  write 

Prolu  the  5  unitu  in   unit**  place  in  the  product  and 

rtivrve  the  tens  to  add  with  the  tens. 

3  timt  -  :ire  21   ten«,  plus  1  ten  n*8er>'e<l  are  22  tenn,  or  2 

hundriHlii  and   2   teni«.     We  write  the  2   tcnA  in  tenx*  place  in  the 

product  and  rewrve  the  hundred«  to  add  with  the  hundredft. 

3  timcfi  2  hundreds  are  0  hundretU,  plujt  2  hundreds  rewrved  arc  8 
hundreds,  which  we  write  in  hundred«'  place  in  the  product.  Hence 
the  produH  i^  HS^,  the  same  as  the  gum  in  the  first  process. 

Proof.  n-sults  obtained  hy  both  processes  agree,  the  work 

is  proh.iblv  corrKi, 

In  multiplying,  pronounce  the  mults  only.  Thus,  in  the  example 
given  above,  instead  of  saying  3  times  5  are  15,  3  times  7  are  21,  plus 
1  reserved  arc  2*2;  3  times  2  are  6,  plus  2  rescr 
22,  8. 

Solve  and  prove: 

2.  3  times  314. 

3.  4  times  50^ 


How  many  are 

8.  5  times  314? 

9.  4  times  655? 
10.  7  times  764? 


..wl        ...... 


4. 

4  times  987. 

6. 

5  times  $819. 

•''• 

5  times  :>45. 

7. 

3  times  $769. 

11. 

3  times  ^30? 

14. 

f^  times  <?42i' 

12. 

6  times  734? 

15. 

6  times  $32? 

13. 

9  times  $48? 

16. 

7  times  $57? 

17.  If  a  man  earns  $17.25  per  week,  how  much  can  he 
<yirn  in  8  weeks? 

18.  A  benevolent  man  paid   annually  ftir  the  support  of 
the  poor  $2365.     How  much  did  lie  pay  in  7  years? 

19.  A  shoe  dealer  sold  9  pairs  of  shoes  at  $3.75  a  pair. 
How  much  did  he  receive  for  all? 

20.  A  man  bought  8  cows  at  an  average  price  of  $31.27. 
How  much  did  thev  all  cost  him? 


MULTIPLICATION.  40 

21.  It'  a  ship  sail  425  miles  in  one  week,  how  far  will  she 
sail  in  9  weeks? 

22.  A  barrel  of  Hour  wi-iglis  lliG  jKninds.     How  mueli  Avill 
8  barrels  weigh? 

23.  When   wheat  is  worth  $1.78  jx^r  bushel,  how  miicli 
can  be  realized  from  the  sale  of  9  bushels? 

24.  At  $6.25  a  pair,  what  will  be  the  cost  of  7  pairs  of 
boots? 

25.  There  are  5280  feet  in  a  mile.     How  many  feet  in  7 
mil( 

20.   Ai  c>7.50  an  acre,  wlmt  will  be  the  cost  of  8  acres  of 
land? 

27.  What  will  be  the  cost  of  7  thousand  feet  of  lumber  at 
$18.25  ixjr  thousand? 

28.  When  broom  corn  is  selling  at  883.50  a  ton  what  is 
the  value  of  8  tons? 

CASE  II. 

77.  When  the  multiplier  is  expressed  by  more  than 
oiif^  flp;ure. 

1.  There  are  9  square  feet  in  one  square  yard.  How 
many  are  there  in  10  square  yards? 

2.  How  many  square  feet  in  6  square  yards? 

3.  Since  10  square  yards  contain  90  square  feet,  and  6 
square  yards  contain  54  .^^quare  feet,  how  may  the  numl)er 
of  square  feel  in  1<»  <>.  <r  10,  square  yards,  l)e  found.^ 
How,  then,  ni;iy  yon  nHihii)ly  l)y  16?     By  18?     By  13? 

4.  Find  the  cost  of  17  yard.^  of  cloth  at  18  cents  a  yard? 

5.  When  eggs  are  21  cents  a  dozen,  what  will  15  dozen 

(•O.St? 

6.  Since  12  inches  make  one  foot  in  length,  how  many 
inches  are  there  in  18  feet? 

7.  A  |K)und  of  sugar  is  equal  to  16  ounces.  How  many 
ounces  are  there  in  a  quantity  of  sugar  weighing  16  iK)unds? 


50  MULTIPLICATION. 

7.  Find  the  cost  of  17  yards  of  cloth  at  8  cents  a  yard, 
by  finding  the  cost  of  9  yards,  and  then  of  8  yards.  Of  10 
yards  and  7  yards. 

8.  What  will  be  the  cost  of  11  primers  at  25  cents  each? 
y.  Find  the  cost  of  16  yards  of  cloth  at  8  cents  a  yard, 

by  finding  the  cost  of  10  yards  and  0  yards.     9  yards  and 
7  yards.     8  yards  and  8  yards. 

10.  James  is  in  school  5  hours  u  day.  How  many  hours 
is  he  in  school  during  three  weeks,  or  15  schooUlays? 

11.  A  ! '  ;i.  lit  }  sets  of  forks,  each  set  containing  6  forks, 
liiiw  niucli  did  the  forks  cost  him  at  $2  each? 

12.  Mary  bought  15  pounds  of  sugar  at  11  cents  a  pound, 
and  3  pounds  of  raisins  at  15  cents  a  pound.  After  paying 
her  bill  she  had  10  cents  lef\.  Ilnw  much  monoy  had  she 
at  first? 

13.  A  cooper  can  make  12  barrels  a  day.  How  many  can 
he  make  in  12  days? 

14.  John  bought  12  lead  pencils  at  8  cents  each,  and  2 
erasers  at  4  cents  each.     How  much  did  all  cost  him? 

15.  The  railroad  fare  from  liochester  to  New  Y(Tk  i>  c7. 
How  much  will  the  tickets  for  a  jwirty  of  9  cost? 

16.  If  a  cow  give  9  quarts  of  milk  a  lay.  how  much  milk 
will  she  give  in  9  da3r8?    . 

17.  If  a  man  put  $8  in  a  savings-bank  each  month,  1m )U 
much  will  he  deposit  in  a  year? 

18.  At  $4  a  yard,  what  will  17  yards  of  hroiukluth  cost? 

19.  If  a  laborer  can  earn  82  a  day,  how  much  can  he  earn 
in  12  days? 

20.  What  will  13  jmirs  of  skates  cost  ai     i  ..  i^uii/ 

21.  At  20  cents  a  dozen,  how  much  will  18  dozen  eggs  cost? 

22.  A  coal  dealer  sold  an  average  of  18  tons  of  coal  per  day 
for  12  days.     How  many  tons  did  he  sell  in  that  time? 

23.  At  22  cents  a  pound,  how  much  will  11  pounds  of 
butter  cost? 


MIM/n  PLICATION. 


51 


24.  How  lar  will  a  man  travel  in  15  days,  if  he  travel 
10  hours  a  day  and  3  miles  an  hour? 

25.  A  man  bought  25  cows  and  12  times  as  many  sheep, 
ilow  many  sheep  did  he  buy? 


WJRITTEN     EXERCISES, 


1st  process. 

327 

123 

1st  Partial  Prod. 

981 

2(1  I»rtrtial  Prod. 

6540 

3d  Piirtlal  Prod. 

32700 

EnUre  Prod. 

40221 

78..  1.  Multiply  327  by  123. 

Analysis. — For  convenience  we  write 
the  numbers  as  in  the  preceding  case. 
Since  in  multiplying  we  must  multiply 
by  the  parts  of  the  multiplier  and  add 
the  partial  products,  to  multiply  by  123 
we  multiply  by  3  units,  2  tens,  and  1 
hundred  as  partial  multipliers. 

3  times  327  are  981,  the  first  partial 
product;  2  times  327  are  654  and  2  tem 
times  327  are  654  tens^  or  6540,  which 
we  write  for  a  second  partial  product. 

1  time  327  equals  327,  and  1  hundred  times  327  are  327  hundreds, 
or  32700,  which  we  write  for  a  third  partial  product.  The  sum  of 
the  partial  products  will  be  the  entire  product. 

Analysis. — In  the  second  process  the 
ciphers  at  the  right  of  the  partial  prod- 
ucts are  omitted,  the  significant  figures 
still  occupying  their  proper  places.  Thus, 
in  multiplying  by  2  tens  the  product  was 
654  tens,  or  6  thousand,  5  hundred,  4 
tens,  which  we  write  in  their  places  in 
the  partial  product. 

In  multiplying  by  hundreds,  the  low- 
est order  of  the  product  is  himdreds, 
hiiuv  we  write  the  first  figure  of- the  product  undtr  hundreds. 

Proof. — Multiply  the  multiplier  by  the  multiplicand.  (Prin.  3.) 
If  the  result  agrees  with  that  formerly  obtained,  the  work  is  probably 
correct. 


2d  process. 

327 

123 

1st  Partial  Prod. 

981 

2.1  Pftrlial  Prod. 

654 

3d  Partial  Prod. 

327 

Entipe  Prod. 

40221 

52 


M  ULTIPLl  CATION. 


Rule. —  Write  the  multiplier  under  the  mvUiplicand  wWi 
uniti  under  unitSy  tens  under  tens,  etc. 

3fuUi})ly  each  figure  oj  the  mvltxplicand  by  eacfi  significant  fig- 
ure of  tlie  vmltiplier  successively,  beginning  iciHi  units.  Place  tlie 
rigid  hand  figure  of  each  product  under  Hie  figure  of  the  multiplier 
used  to  obtain  it,  and  add  the  partial  products. 

Proof. — Review  the  work,  or  midtiply  the  multipUcr  by  the 
multiplicand.     If  Die  restdts  agree  the  uxyrk  is  j)robably  correct. 


EXAMPLES. 

(2.) 

(3.) 

(4.) 

(5.)     (6.) 

Multipl 

y  325 

219 

384 

$2.81    $3.18 

By 

42 

54 

46 

23      36 

Multipl 

y: 

Multiply: 

7. 

456 

by  12. 

23. 

73982  by  321. 

8. 

389 

by  23. 

24. 

42586  by  604 

9. 

493 

by  25. 

25. 

89258  by  703. 

10. 

374 

by  27. 

26. 

84206  by  569. 

11. 

3625 

by  28. 

27. 

156783  by  423. 

12. 

2413 

by  31. 

28. 

248164  by  372. 

13. 

3681 

by  63. 

29. 

182642  by  419. 

14. 

67021 

by  52. 

30. 

192573  by  429. 

15. 

63583 

by  62. 

31. 

234567  by  612. 

16. 

84216 

by  78. 

32. 

467105  by  623. 

17. 

38413 

by  35. 

33. 

398120  by  706. 

18. 

29615 

by  45. 

34. 

683912  by  1684. 

19. 

23423 

by  25. 

35. 

312465  by  1827. 

20. 

24542 

by  64. 

36. 

468975  by  2946. 

21. 

45684 

by  73. 

37. 

416004  by  3009. 

22. 

41075  by  62. 

38. 

329706  by  3802. 

MULTIPLICATION.  53 


39.  8  18.61  by  73. 

40.  §115.81   by  45. 

41.  $164.32  by  81. 

42.  $123.45  by  804. 

43.  $415.05  by  367. 


44.  $  18.37  by  127. 

45.  $113.41  by  613. 

46.  $281.69  ])y  247. 

47.  $312.09  by  684. 

48.  $425.27  by  618. 


49.  In  a  reaper  factory  an  average  of  2346  reapers  is  con- 
structed annually.  At  this  rate  how  many  would  be  made 
in  25  years? 

50.  A  farmer  counted  the  trees  in  his  orchard  and  found 
that  he  had  104  rows,  each  row  containing  106  trees.  How 
many  trees  were  there  in  the  orchard? 

51.  In  a  croquet  factory  a  man  makes  835  balls  daily. 
How  many  balls  Ciin  he  make  in  312  days? 

52.  The  distance  between  Rochester  and  Syracuse  is  81 
miles.  How  many  miles  per  month  of  31  days,  will  a  loco- 
motive travel  that  goes  from  Rochester  to  Syracuse  daily  and 
returns? 

53.  Mr.  Davis  built  8  houses  at  a  cost  of  $1925  each 
6  at  $227.".  o:u'h.  and  5  at  $3897  each.     What  did  they  all 
cost  him 

54.  Sold  my  farm  of  413  acres  at  $85  per  acre.  How 
much  did  I  get  for  it? 

55.  How  much  will  it  cost  to  build  89  miles  of  railroad 
at  an  estimated  expense  of  $57394  per  mile? 

CASE  III. 

79.  Wliorc  there  are  ciphers  on  the  right  of  either 
or  both  TuctorM. 

1.  How  many  are  are  10  times  2?     3?     4?     5?     6?     7? 
8?    9? 

2.  Write  the  ulxivc  mulliplicamU  ami   products  side  by 
side  and  compjire  them. 


64  MULTIPLICATION. 

3.  How  may  the  product  be  iound  iVom  the  multipHoand 
when  the  multiplier  is  10  ? 

4.  How  many  are  100  times  '1'.  '■>:  \:  '>':  •;/  7/ 
8?    9? 

5.  How  may  any  number  be  multiplied  by  100?  by  1000? 
(5.  How  may  a  numl)er  be  multiplied  by  1  with  any  num- 
ber of  ciphers  affixed  ? 

80.  Principle. — In  muUipltfitig  by  10,  100 ,  1000 y  etc,  as 
many  cipfiers  must  be  annexed  to  the  right  of  the  multiplicand  as 
there  are  ciphers  in  ilie  multiplier. 

1.  Multiply  36  by  1000. 

PROCESS.  Analysis. — Since  in  muhiplving  hy  1  with    any 

3  6  number  of    ciphers   annexed,    we   annex    as    many 

1000  ciphers  to  the  multiplicand  as  there  are  in  the  mul- 

tiplier,  to  multiply  by  1000  we  annex  three  ciphers 

3  6  000  to  x\\Q  multiplicand, which  gives  the  product  36000. 

2.  Multiply  2360  by  400. 

PROCESS.  Analysis. — Since  2360  is  equal  to  236  X  10,  and 

2360  ^^  "  ^""*  to  4  X  100,  the  product  of  2360  X  400  may 

±(\C\        ^  obtained  by  multiplying  2^36  by  4,  and  this  product 

1ji!L      by  10  times  100,  or  1000.     The  product  of  236  X  4  is 

9  44000      944,  and  this  may  bs  multiplied  by  1000  by  annexing 
three  ciphers  (Prin.),  giving  as  :\  result  944000. 

KuLii. — Multiply  witliout  reyaid  to  the  ciphers  on  the  right, 
ami  to  the  product  annex  as  many  cipliers  as  there  are  on  tlie 
right  of  both  multiplier  and  multiplicand. 

3.  Multiply    375  by    10. 

4.  Multiply    845  by    30. 

5.  MultiplV    176  by  500. 

6.  ]Multiply  1385  by  200. 

7.  Multiply  4860  by  250. 
8  Multiply  3120  by  210. 


By    100. 

By      40. 

By    300. 

By      70. 

Bv    600. 

By    900. 

By    700. 

Bv    400. 

By  1000. 

Bv  2000. 

Bv  2200. 

By  3300. 

By  3200. 

By  4200. 

By  6500. 

By  3800. 

By  2700. 

By  4600. 

Mill  ll'l.lCATIOX.  55 

9.  In  a  mile  there  are  5280  feet.  J  low  many  leet  are 
there  in  500  miles? 

10.  In  an  acre   there   are   160  square  rods.     How  many 
square  rods  are  there  in  a  farm  of  300  acres? 

11.  A  farmer  sold  a  flock  of  260  sheep  at  $3.20  per  head. 
How  much  did  he  jjet  for  them  ? 

12.  A  drover  sold  1120  hogs  at  an  average  price  of  $16.30 
per  head.     How  much  did  he  receive  for  them? 

EXAMPLES. 

81.    1.  What  will  be  the  cost  of  896  chests  of  tea,  each 
chest  containing  58  pounds,  at  63  cents  a  pound? 

2.  An  agent  sold  3923  Lyman's  Historical  Charts  at  $3.50 
each.     How  much  did  he  receive  for  them? 

3.  I  have  6  bins  that  hold  119  bushels  each.  They  are 
full  of  gniin  and  I  have  already  sold  515  bushels.  It  was  all 
raised  on  my  farm  this  year.     How  much  grain  was  raised? 

4.  A.  J.  Newton  &  Co.  bought  113  cases  of  calico,  each 
case  containing  64  pieces,  and  each  piece  47  yards.  How 
many  yards  did  they  buy? 

5.  A  drover  bought  25  oxen  at  $85  a  head,*  316  sheep  at 
84.50  a  head,  and  94  calves  at  $8  a  head.  What  was  the 
whole  amount  paid? 

6.  A  man  insured  2  houses  valued  at  $3750  and  $4650, 
respectively,  at  the  rate  of  $2  j)er  hundred  dollni-s.  How 
much  did  the  insurance  cost  him? 

7.  If  I  have  219  acres  of  land,  and  each  acre  produces 
47  bushels  of  corn,  how  many  bushels  do  I  receive? 

8.  How  many  quills  can  be  obtained  from  398  geese,  if 
each  wing  furnishes  6  quills? 

9.  A  grocer  sold  in  one  month  81  dozen  eggs  at  26  cents 
per  dozen ;  in  the  next,  53  dozen  at  28  cents  per  dozen.  How 
much  money  did  he  receive  for  the  eggs? 


56  MI    i.ril'I.K   A  I  ION. 

in.  It  riMjiiirt-  17  hi  picki-is  to  fciic'  mim-  .;,!,•  ,,t'  a  >(|uare 
lot.  How  many  pickets  will  be  recjuirtd  to  Icn.  ,  1.;  |,,!>  >>[' 
the  .-am;'  .-!/.»•  and  shape? 

11.  A  sold  1.;  firkins  of  butter,  each  firkin  contiiininj  Id 
pound-,  at  >  .."1 1  a  i.ound.     How  much  did  he  receive  for  it  ? 

12.  A  eoal  dealt-r  Unight  13  car  loads  of  coal,  eacli  load 
containing  10  tons,  at  8G.85  a  ton.  He  retailed  48  tons  of 
this  at  87  per  ton,  28  tons  at  88.25  per  t-n.  1'7  t  iis  at  88.75 
per  ton,  and  tlio  remainder  at  89.50  per  t.  n.  How  much  did 
he  make  I>\  tlio  transaction? 

!.'>.  An  army  lo>t  in  hattle  315  killed,  417  wounded;  the 
enemy  Iom  in  kill-  !  ■  '  oimded,  together,  13  times  as  many. 
How  maiiN  -  Idi.  :  i lied  and  wounded  in  this  battle? 

11.  It  two  steamers  should  leave  New  York  at  the  same 
tinn  ,  an  I  J  ...«l,]  <.y^\  j^  tlie  same  direction,  the  first  at  the 
rail   ot    I  n  hour,  the  second  at  the  rate  of  15  miles 

an  lioni-.  Ix-u  lar  apart  would  they  be  in  36  hours? 

1").  .Mr.  Ilnd^on  houLdit  'llO  bushels  of  corn  at  65  cents  a 
Nu-licL  L'lo  Imi-Ii.I.  nf  uhrat  at  81.35  i)er  bushel,  and  273 
bushels  of  iiai-  at  f!  rriils  a  hii-^hcl.  What  did  the  whole 
cost  111  Ml? 

lf>.  Ml .  1 !  M  i(  rson  sold  a  farm  of  325  acres  at  865. oO  p(  r 
acre,  and  received  in  i)ayment  345  sheep  at  83.25  per  head,  a 
note  for  82684.95  and  tlu'  rest  in  cash.     Plow  muHi  cash  did 

llr    nvrive? 

17.  .V  clotli  merchant  sold  two  lots  of  cassimeres,  the  first 
coniaiiiinL^  17  pieces  of  2'H  yards  each,  at  81.75  per  yard,  the 
second  coniainini:-  '2'-)  pi.c*-  aNira-ing  29  yards  each,  at  81.85 
j)er  yard.      Wliat  was  the  value  of  the  whole? 

IS.  An  excursion  train  comoosed  of  13  passenger  coaches, 
etich  containinu-  .".7  j:  !:T   from  Syracuse  to  Niagara 

Falls  and  back.  If  the  tiue  to  Niagara  Falls  and  return  to 
Syracuse,  was  83.25  per  ticket,  how  much  did  the  railroad 
com|Kiny  receive? 


IN1>VCTIVE    EXERCISES, 

82.    1.  How  many  groups  of  2  birds  each  can  be  formed 
from  6  birds?     How  many  2's  are  there  in  6? 

2.  How  many  groups  of  3  sheep  each  can  be  formed 
from  9  slieep?     How  many  3's  are  there  in  9? 

3.  How  many  groups  of  2  chickens  each  can  be  formed 
from  10  chickens?    How  many  2*8  are  there  in  10? 

4.  At  5  cents  apiece,  how  many  pencils  can  be  bought 
for  10  cents?     How  many  5's  are  there  in  10? 

5.  AVlien  milk  is  worth  7  cents  a  quart,  how  many  quarts 
can  be  bought  for  28  cents?     How  many  7's  are  there  in  28? 

6.  There  are  20  panes  of  glass  in  the  front  of  a  block  of 
istores.  If  each  window  contains  4  panes,  how  many  win- 
dows arc  there?     How  many  4's  are  there  in  20? 

7.  At  8  cents  a  dozen,  how  many  peaches  can  be  bought 
for  24  cents?  How  many  times  8  cents  are  24  cents?  How 
many  8's  are  there  in  24? 

8.  How  many  groups  of  4  things  each  can  be  formed 
from  ^(^  tilings?     How  many  4's  are  there  in  16? 

'.  A  merchant  had  30  yards  of  calico  which  he  cut  into 
j)ieces  T)  yards  long.  How  many  pieces  did  it  make?  How 
many  o's  are  there  in  30?  How  many  times  is  5  contained 
in  r>0? 

10.   How  many  ^'s  are  there  in   1'^/      ll<iu   many  limes  is 
9  contained  in  18? 

(.-,7) 


58  DIVISION. 

11.  lIi)W  juaiiv  ."»"s  Mrc  thon-  in  lU.'  in  \')'f  In  '20?  In 
25?    Ill  30? 

12.  If  15  cents  are  divided  equally  among  3  boys,  how 
many  cents  will  each  receive?  When  15  cents  are  divided 
into  3  equal  j)arts,  how  many  cents  will  each  part  contain? 

13.  If  12  i^eaches  are  arranged  in  3  rows,  how  many  will 
there  be  m  each  row? 

14.  AVhat  is  one  of  the  4  equal  parts  of  8?   Of  12?    Of  16? 

15.  How  many  3*8  are  there  in  30?  How  many  are  10 
threes,  or  10  times  3? 

16.  How  many  4*8  are  there  in  40?  How  many  times  is 
4  contained  in  40?    How  many  are  10  fours? 

DEFINITIONg. 

83.  Division  iri  the  process  of  finding  how  many  times 
one  number  is  contained  \u  another;  or, 

The  process  of  separating  a  number  into  equal  parts. 

84.  The  Dividend  is  the  number  to  be  divided. 

85.  The  Divisor  is  the  irumber  by  which  we  divide. 
It  shows  into  how  many  equal  parts  the  dividciul  i«  to  be 
divided. 

86.  The  Quotient  is  the  result  obtained  by  division. 
It  shows  how  many  times  the  divisor  is  contained  in  the 
dividend. 

87.  The  jiart  of  the  dividend  remaining  when  the  division 
is  not  exact  is  called  the  Remainder, 

88.  The  Sign  of  Division  is  ^-  .  It  is  read  divided 
by.  AVhen  placed  lietween  two  numbers  it  shows  that  the 
one  at  the  left  is  to  be  divided  by  the  one  at  the  right. 

Thus,  154-^7,  is  read  154  divided  by  7. 


I)l\  IMO.N.  59 

Division  is  also  indicated  by  j)lacing  the  dividend  above 
the  divisor  with  a  line  between  them,  and  by  writing  the 
divisor  at  the  leil  of  the  dividend  with  a  curved  line  between 
them.  Thus,  154  divided  hy  7,  may  also  bo  written  ^5"*,  and 
7J154. 

89.  Principles. — 1.  The  dividend  and  divisor  mud  he  like 
numbers. 

2.  The  quotient  vimt  be  an  abstract  number. 

3.  The  product  of  the  divisor  by  Hie  qiiotientf  plus  the  remain- 
dery  is  equal  to  tlie  dividend. 

«)  1.  In  problems  where  It  is  required  to  separate  a  ninnlxr 

;;        into  equal  parts,  it  \n  customary  to  regard  the  dividend  and 

—       quotient    as   like   numbers,    and    the   divisor   as   an    abstract 
^       number. 

J_  2.  The  example  "How  many  3's  are  there  in  9"  may  be 

3        solved,  as  in  the  margin,  by  mbtradion.     All  examples  in  divis- 

3        ion  may  lie  solved  in  the  same  manner.     Hence,  division  may 

be  regarded  as  a  short  method  of  mhtmctbiy  eqwd  numbers. 

3.  In  multiplication  two  numbers  are  given  to  fiud  their  product 

In  division  the  product  is  given  and  one  of  the  factors  to  find  the  other. 

Hence,  division  is  the  converse  of  multiplication. 

CASE  I. 

90.  UTieii  the  clivinor  is  expressoil  by  one  f]g;ure. 

1.  At  >?S  each, how  many  plows  can  be  bought  for  824? 

Analysis. — Since  each  plow  costs  $8,  as  many  plows  can  be  bought 
for  $24  as  $8  is  contained  times  in  $24,  which  is  3  times.  Therefore 
3  plows  can  be  bought  for  $24. 

2.  If  a  man  can  earn  ?7  in  a  day,  how  long  will  it  tiike 
him  to  earn  828? 

3.  At  §4  each,  how  many  hats  can  be  bought  for  824? 

4.  When  flour  is  selling  at  86  a  hundred-weight,  how  nutuy 
hundred-weight  can  be  bought  for  836? 


60  DIVISION. 

.).    ir  a  Miiison  built  3  rods  of  walk  per  (lay,  how  loner  (\u\ 
it  take  him  to  build  21  rods? 

6.  B  imid   96  cents  for  glass  at  8  cents  a  pane.     How 
many  panes  did  lie  buy? 

7.  At  $d  a  cord,  how  many  cords  of  wood  can  be  bought 
for  $45? 

8.  If  a  man  earns  Sll  a  week,  how  many  weeks  will  he 
require  to  earn  $66? 

9.  How  many  lots  ot    1  i  :i<  i-  <  each  can  Ixi  made  from  a 
farm  containing  132  acres? 

10.  If  a  farmer  exchanges  6  firkins  of  butter  worth  $20  a 
firkin  for  cloth  at  $4  a  yard,  how  many  yards  will  he  receive? 

11.  Aly  coal  cost  me  $35  at  tlie  rate  of  $7  a  ton.     How 
many  tons  did  I  purchase? 

12.  How  many  engravings  must  an  artist  h  11  Im  >\2  apiece 
to  realize  $84? 

13.  When   sugar  is  worth   9  cents  a  pound,  how  many 
pounds  can  be  bought  for  45  cents? 

14;  At  the  rate  of  $7  a  rod,  how  many  rods  of  fence  can 
be  built  for  $63? 

15.  I  hired  a  man  for  $45  to  tl«>  a  pirce  of  work  at  tho  rate 
of  $5  a  day.     How  many  days  did  it  take  him? 

16.  A  lady  bought  some  silk  worth  S3  a  yard,  paying  $36 
for  it.     How  many  yards  did  she  buy? 

17.  How  many  barrels  of  flour  at  $8  a  barrel  can  be 
l)()ught  for  $48?  ' 

18.  How  many  jK)unds  of  nails  can  he  bought  for  75  cents 
at  the  rate  of  4  pounds  for  20  cents? 

19.  I  lx)ught  6  sheep  for  $30.     How  much  did  I  pay  per 
head  ? 

20.  At  $5  i^er  head,  how  many  head  of  sheep  can  be  bought 
for  $37?  Am.  7  sheep  and  $2  left. 

21.  A  man  whose  wages  were  $4  a  day  earned  in  a  certain 
time  $33.    How  many  days  did  he  work?        Am.  8 J  days. 


DIVISION.  61 

91.  1  loiii  examples  20  and  21  it  is  apparent  that  the 
remainder  may  be  written  either  after  the  quotient,  as  in  the 
answer  to  the  20th,  or  as  a  part  of  it,  as  in  the  answer  to 
the  21st. 

When  written  as  a  part  of  the  quotient,  the  remainder  is 
expressed  by  placing  the  divisoi'  wider  it  with  a  line  between 
them.  Such  an  expression  shows  that  each  unit  of  the  re- 
mainder is  to  be  divided  into  as  many  equal  parts  as  there 
ftre  units  in  the  divisor. 

.When  any  thing  is  divided  into  tvx>  equal  parts,  each  of  the 
parts  is  called  oiie.  half. 

When  into  three  equal  parts,  each  part  is  called  one  third. 

When  into /o«r  equal  parts,  each  part  is  called  one  fourth. 

When  into  five,  six,  seven,  etc.,  equal  parts,  the  parts  are 
called  fifths,  sixths,  sevenths,  etc. 

^  expresses  om  half,  or  one  of  two  equal  parts  of  any 
thing. 

\  expresses  one  fourth,  or  one  of  four  equal  parts  of  any 
thing. 

J  expresses  two  fifty,  or  two  of  five  equal  parts  of  any 
thin^:. 

Tj^-  expresses  Jive  iivoity-sevenths,  or  five  i.i  twenty-seven 
equal  parts  of  any  thing. 

92.  One  or  more  of  the  equal  parts  of  any  thing  is  called 
I  Fraction. 

93.  Read  the  following  fractional  expressions: 

¥  inr  TT  T5  M  tI 

84  T9  "nnr  tt  W  It 

22.  If  James  should  divide  25  apples  equally  among  5 
boys,  what  part  of  the  whole  would  each  receive?  How 
many  apples  would  each  receive? 


62 


DIVISION. 


Analysis. — I!  he  should  divide  25  apples  equally  among  5  boys, 
each  boy  would  receive  (me-jifth  of  25  apples,  which  is  5  apples. 

23.  If  flour  is  worth  $8  a  barrel,  what  will  one-half  barrel 
cost? 

24.  Mr  Smith  bought  8  bushels  of  chestnuts  for  824. 
How  much  did  he  pay  per  bushel?  How  much  is  one-eighth 
of  24? 

25.  What  is  one-sixth      of  36? 

26.  What  is  one-tenth      of  50? 

27.  What  is  one-seventh  of  14? 


Of  42? 

Of  48? 

Of  60? 

Of  60? 

Of  70? 

Of  80? 

Of  28? 

Of  42? 

Of  49? 

^'J^JTT£X    EXERCISES. 


94.    1.  Divide  1396  by  4. 


IST  PROCESS. 
Divisor.  Dividend.  Quotient. 

4)1396(300  times. 
1200      40  times. 


196 
160 

36 

9  times. 
349  times. 

36 

2d  process,  g  S  1 
4)1396(349 
12 

19 

16 

36 
36 

the  partial  dividend,  there  is 


Analysis. — For  convenience  we 
write  the  divisor  at  the  left,  and  the 
quotient  at  the  right  of  the  dividend, 
with  curved  lines  between  them,  and 
begin  at  the  left  to  divide. 

4  is  not  contained  in  1  thousand 
any  thousand  times,  therefore  the 
quotient  can  not  contain  units  of  any 
order  higher  than  hundreds.  Hence 
we  find  how  many  times  4  is  con- 
tainc<l  in  all  the  hundreds  of  the 
dividend.  1  thousand  plus  3  hun- 
dreds equals  13  hundreds.  4  is  con- 
tained in  13  '  hundreds  3  hundred 
times  and  a  remainder.  We  write 
the  3  hundreds  in  the  quotient  and 
multiply  the  divisor  by  it,  obtaining 
for  a  product  12  hundreds,  or  1  thou- 
sand 2  hundred,  which  we  write  under 
units  of  the  same  order  in  the  divi- 
dend. Subtracting  this  product  from 
a  remainder  of  1  hundred. 


DIVISION.  63 

1  hundred  plus  9  tens  equals  19  tens.  4  is  contained  in  19  tens  4 
tens  times  and  ti  remainder.  We  write  the  4  tens  in  the  quotient  and 
multiply  the  diviiior  by  it,  obtaining  for  a  product  16  tens,  or  1  hundred, 
and  0  tens,  which  we  write  under  units  of  the  rame  order  in  the  partial 
dividend.    Subtracting,  there  is  a  remainder  of  3  tens  and  6  units. 

3  tens  plus  G  units  equals  36  units.  4  is  contained  in  36  units  9 
timer..  We  write  the  9  units  in  the  quotient  and  multiply  the  divisor 
by  it,  obtaining  for  a  product  36  units,  or  3  tens  and  6  units,  which  we 
write  under  units  of  the  same  order  in  the  partial  dividend.  Subtract- 
ing, there  is  no  remainder.     Hence  the  quotient  is  349. 

In  the  second  process  all  ciphers  are  omitted  from  the  right  of  the 
products  and  the  significant  figures  are  written  under  units  of  the  same 
order.  The  (juotient  also  is  expressed  by  writing  the  different  orders  of 
units  in  proper  succession. 

Proof. — 349  the  quotient,  multiplied  by  4  the  divisor,  is  equal  to 
1396  the  dividend. 

Hence  the  work  is  correct.     (Prin.  3.) 


Solve  in  like  manner  and  prove : 


2.  738 --3. 

3.  845-^5. 

4.  385 --7. 


5.  4821 --3. 

6.  3462 -V- 6. 

7.  3864 -f- 8. 


8.  7848-^9. 

9.  8432 --4. 
10.  8308 --7. 


95.  The  solution  of  the  preceding  examples  may  be  short- 
ened hy  performing  the  multiplications  and  subtractions  with- 
out writing  the  results.  This  process  is  called  Short 
Division, 

The  solution  of  Example  1  by  Short  Division  is  as  follows: 

PROCESS.  Analysis. — 4  is  contained  in  13  hundred  3 

4")  139tj  Imndred  times  and  I  hundred  remainder.     We 

write  3  hundreds  in  the  quotient  under  units 

Quotient        3  49  ^^  ^j^^  ^^^^^  ^^^,^r  in  the  dividend.     1  hundrtd 

remainder  united  with  9  tens  makes  19  tens. 
4  is  cuutaiiiKl  in  VJ  lens  4  tens  times  and  3  tens  remainder.  We  write 
the  4  tens  in  the  quotient  under  tens  of  the  dividend.  3  tens  remain- 
der unitetl  with  6  units  make  36  units.  4  is  contained  in  36  units  9 
times.     We  write  the  9  in  the  quotient.     Henee  the  quotient  is  349. 


64 


J>1  VISION. 


Solve  by  short  divUUm: 

11.  4872 -V- 4.  17.  4^. 

12.  0830-^5.  18.  ^f^. 

13.  2976  —  6.  19.  ^. 

14.  2985-4-5.  20.  ^. 

15.  4635 -r- 3.  21.  4^«. 

16.  3936 -f- 4.  22.  £^. 


23.  4567       . 

24.  8932  :  (5. 

25.  8174-9. 

26.  9185 --4. 

27.  8436 -- 7. 

28.  3885 -T- 8. 


CASE  U. 

\H'i.  \Vhvn  the*  cli%lMor  is  expreiMed  bj  niori*  tliiiii 
ouc>  ilKiiro. 

1.  How  niauy  barrels  of  flour  at  $10  a  barrel  can  be  bought 
for  $80? 

Analysis. — Since  1  barrel  ooRtM  $10,  an  many  barrels  can  be  bought 
for  $80  as  $10  arc  contained  times  in  $80  which  is  8  times.  Therefore 
8  barrels  can  be  bought  for  $80. 

2.  How  many  pounds  of  mutton  at  10  cents  a  pound  can 
be  bought  for  50  cents?  How  many  lO's  in  50?  In  60: 
Li  70?    In  80? 

3.  A  man  measured  a  .<tick  and  found  it  to  be  60  inche? 
long.  There  are  12  inches  in  a  foot.  How  many  feet  long 
was  it?     How  many  12's  in  60?     In  72?     In  84?    In  96? 

4.  At  $13  a  ton  how  much  hay  can  be  bouglit  for  $26? 

5.  At  15  cents  each  how  many  toys  can  be  bought  for  30 
cents?     For  45  cents?     For  60  cents? 

6.  Mr.  Henderson  sold  20  lambs  for  880.  How  much 
did  he  get  apiece  for  them  / 

7.  25  cents  make  a  quarter  of  a  dollar.  How  many 
quarters  of  a  dollar  has  a  boy  who  luis  50  cents? 

8.  Henry's  father  gave  him  a  dollar.  How  many  pine- 
apples at  20  cents  each  can  he  buy  with  the  money? 

9.  The  railroad  flire  to  a  certain  place  is  35  cents.  How 
many  tickets  can  be  bought  with  70  cents? 


DIVISION.  ,    <;  > 

10.  If  a  lM)y  earns  11  cents  an  hour,  how  long  will  it  take 
him  to  earn  55  cents?    66  cents?  '  88  cents? 

11.  There  are  20  hundred-weight  in  a  ton.     How  many 
tons  are  there  in  45  hundred-weiglit?     How  many  in  55? 

12.  In  one  day  there  are  24  hours.     How  many  diiys  are 
there  in  50  hours?     In  60  hours?     In  72  hours? 

13.  12  articles  make  a  dozen.     How  many  dozen  arc  there 
in  39  articles?    In  48?     In  51?    In  60?    In  65? 

14.  A  farm  of  60  acres  was  divided  into  15  equal   lots. 
How  many  acres  were  there  in  each  lot? 

15.  At  18  cents  a  dozen,  how  many  dozen  of  eggs  can  be 
bought  for  36  cents?    For  40  cents?     For  54  cents? 

16.  When  butter  is  30  cents  a  pound,  how  many  ix)unds 
can  be  bought  for  90  cents?    For  81.20?     For  61.50? 

17.  How  many  20's  are  there  in  40?    In  50?    In  60? 

18.  How  many  25's  are  there  in  50?    In  60?    In  75? 

19.  By  selling  brooms  at  25  cents  each,  I  received  $1.25. 
How  many  brooms  did  I  sell? 

20.  In  100  how  many  lO's  are  there?    How  many  ITs? 
12'8?    13's?    15's?    16's?    20's?    23's?    25's? 


WRITTEN     EXERCISES, 

97.    1.  Divide  7975  by  26. 

PROCESS.  Analysis. — 26  is  not  contained  in  7 

Divisor.  Dividend.  Quotient  thousands  any  thousands    times;   hence 

26)7975(306i!i4        ^'*^  unite  the  thou.sands  with   the  hun- 

iTo  dreds,  making  79  hundreds.     26  is  con- 

tained  in  79  hundreds  3  hundred   times 

1  •  '^  and  a  remainder.     We  write  the  3  hun- 

15  6  drcds  in  the  quotient  and  muhiply  tlie 

\  9  divi.sor  hy  it,  obtaining  for  a  prochict  78 

hundred.s,  or  7  thousands   and    8  hun- 

ilre<ls,  wliich  we  write  under  units  of  the  same  order  in  the  dividend. 

Siilttr.ictin.v   tli.r.'  !<    ;i    rini;iiiw].r   nf  1    liiiiulnd. 


T>I  VISION. 

We  iiniir  iidnil  wiili  iln  making  17  tens.     2^  U 

not  contained  in  17  lens  any  t^ns  times;  therefore  there  are  : 
in  the  quotient,  and  we  write  a  cipher  there. 

Wc  unite  the  17  tens  with  the  5  units,  making  175  units 
foiitaineci  in  175  units  6  times  and  a  remainder.  We  write  tiu-  u  m 
units'  place  in  the  quotient  and  multiply  the  divisor  by  it,  obtain- 
ing for  a  pioduct  150  units,  or  1  iiundred,  5  tens,  and  6  units,  which 
we  write  under  units  of  same  order  in  thr  pnrtinl  dividend.  Sub- 
tracring,  there  is  a  remainder  of  11'.     ^^  naindcr  over 

the  divisor  as  a  part  of  tiic  quotient. 

Hence  the  quotient  b  306^ |. 

Proof.— 306  X  26  -f  19  ' '  nee  the  work  is  correct.  (Prin.  8.) 

f)S.  When  the  steps  in  the  Holution.of  an  example  in  divis- 
ion are  written,  the  process  is  called  IjOng  Ui  vis  ion. 

Rule. — Write  tlie  divi^r  ni  ihr  Iff  nf  ihr  iHviihud  icWi  a 
curved  line  between  them. 

Find  how  many  time:*  the  divinor  in  coutaintd  in  the  feived 
figures  on  the  lejl  hand  of  the  diridcnd  Ihnf  will  mnfaJn    if,    mid 

tcrite  Hie  quotient  on  the  right. 

Mii^''        "     livitor  by  '  ndjiUiecf.  f  laaUr 

the  fiiji  I'-d.     Subti  '  from  the  .      '       dividend 

xisedy  and  to  die  remainder  annex  the  next  figure  ef  the  dividend. 

Divide  "    ■  nitil  all  the  figures  of  the  dividend  have  been 

annexed  to  ^  nder. 

If  any  partial  dividend  mil  not  contain  Hie  divisor,  write  a 
ri}}lier  in  the  quotient,  then  annex  •'  '  Hhe  dividend 

and  proceed  as  before. 

If  tJtere  is  a  retnainder  after  the  lad  division  tvrite  it  after  the" 
qtiotient,  or  wnVA  the  diviior  under  it  as  part  of  tlie  quotieid. 

Proof. — Multiply  the  divisor  by  the  quotient,  and  to  the  prml- 
vct  add  the  remainder,  if  any.  If  the  work  is  correct,  the  result 
will  equal  the  dividend. 

1.  To  find  the  quotient  figure,  see  how  many  times  the  fir^tfif/ure  of  the 
divisor  is  contained  in  fiitU  fi(/u res  oi  the  partial  tlividt'iul  that  will  <-oii- 


DIVISION. 


67 


taiti  it,  making  allowance  for  the  addition  of  the  tens  from  the  pro<l- 
uct  of  the  second  figure  of  the  divisor. 

2.  If  the  product  of  the  <livisor  hy  the  quotient  figure  be  greater 
than  the  partial  dividend  from  which  it  is  to  be  subtracted,  the  quo- 
tient figure  is  too  larrje, 

3.  Each  remainder  must  be  less  than  the  divisor;  otherwise  the 
quotient  figure  is  too  smalf. 

4.  Wlien  there  is  no  remainder  the  divisor  is  said  to  be  exacL 


EXAMPLES. 

99.  Divide: 

Divide 

2. 

1240  by  10. 

25. 

12456  by  24. 

3. 

3443  by  11. 

26. 

28350  by  54. 

4. 

2592  by  12. 

27. 

50854  l)y  94. 

5. 

3978  by  13. 

28. 

58176  by  96. 

6. 

5684  by  14. 

29. 

56394  by  78. 

7. 

6480  by  15. 

30. 

54944  by  101. 

8. 

8736  by  21. 

31. 

90992  bv  121. 

9. 

1472  by  32. 

32. 

199864  by  301. 

10. 

9672  by  31. 

33. 

475524  by  612. 

11. 

9724  by  22. 

34. 

1445204  by  802. 

12. 

2952  by  72. 

35. 

1760225  by  905. 

18. 

1188  by  54. 

36. 

3156584  by  722. 

14. 

4235  by  55. 

37. 

5173302  by  834. 

15. 

5356  bv  52. 

38. 

5926431  by  643. 

16. 

8733  by  41. 

39. 

3214664  by  566. 

17. 

9639  by  81. 

40. 

6923471  by  555. 

18. 

7991  by  61. 

41. 

14293624  by  675. 

19. 

2508  by  22. 

42. 

56243121  by  686. 

20. 

7332  by  52. 

43. 

692348726  by  897. 

21. 

4824  by  72. 

44. 

496839715  by  1047. 

22. 

16665  by  33. 

45. 

786935846  by  3118. 

23. 

13545  by  43. 

46. 

1234589640  bv  96813 

24. 

25578  by  63. 

47.  31964875932  by  37425 

68  DIVISION. 

48.  Into  how  many  lots  of  39  acres  each,  can  a  tract  of 
land  containing  G318  acres  be  dividetl? 

49.  Wm.  Wallace  has  17  horses,  the  aggregate  vamc  ot 
which  is  84^386.     What  is  the  average  worth  of  each  horse? 

50.  ^  surveyor  traveled^  4160(i  rods  in  one  week.  How 
many  miles  did  he  travel,  there  being  320  rods  in  a  mile? 

51.  How  many  eggs  at  8.38  per  dozen  fmn  be  bought  for 
«6.84? 

52.  In  24  hours  the  eaiiu  iiii»\i>  i.ii./wtM*  iniles.  How  far 
does  it  move  in  one  minute,  GO  minutes  making  an  hour? 

53.  Mount  Everest  in  Asia  is  29100  feet  high.  There  are 
5280  feet  in  a  mile.     How  many  miles  high  is  it? 

54.  It  required  4375480  bricks  to  build  an  orphan  asylum. 
How  many  days  did  it  require  5  teams  to  draw  the  bricks,  if 
they  drew  5  loads  per  day  and  1250  bricks  at  a  load? 

55.  The  earth  is  91500000  miles  from  the  sun.  How  many 
seconds  does  it  take  light  to  come  from  the  sun  to  the  earth, 
if  it  travels  185000  miles  per  second? 

56.  A  man  bought  a  farm  of  278  acres  at  $63  an  acre. 
He  paid  81275  down  and  agreed  to  pay  the  rest  in  8 
equal  annual  payments.  How  much  was  he  refinircd  to  pay 
yearly  ? 

57.  The  earnings  of  a  certain  railroad  were  83G81452  during 
the  year.  The  number  of  days  in  a  year  is  3G5.  What  was 
the  average  income  |)er  day? 

58.  How  many  feet  are  there  in  a  mile,  if  42  miles  contain 
221760  feet? 

59.  If  the  average  wages  of  a  laboring  man  are  8500  per 
year,  how  many  men  will  it  require  to  earn  850000  per  year, 
the  salary  of  the  President? 

60.  The  area  of  the  State  of  North  Carolma  is  50704  square 
miles,  and  the  jwpulation,  according  to  the  census  of  1870, 
was  1071400.  How  many  persons,  on  an  average,  were  there 
living  on  a  square  mile? 


uivibiu-N.  69 

CASE  III. 

100.  H'hen  the  <liviN4>r  Iihm  eiphern  on  the  rig^lit. 

1.  How  many  lO's  and  what  remainder  in  46?     84?     97? 

2.  When  a  number  is  divided  by  10,  what  part  is  re- 
mainder?    AVhat  part  is  quotient? 

3.  How  many  hundreds,  and  what  remainder  in  434?  516? 
639?     758?  •  . 

4.  When  a  number  is  divided  by  100  what  part  of  it  is 
remainder?     AVhat  part  is  quotient? 

5.  When  a  numl)er  is  divided  by  1000,  what  jmrt  is  re- 
mainder?    What  part  is  quotient? 

101.  PRixcrrLE.— /??.  dividing  by  10,  100,  1000,  etc.,  the 
remainder  will  he  as  many  of  the  figures  at  the  right  of  the  divi- 
dend as  iliere  are  ciphers  on  iJie  right  of  Hie  divisor.  The  rest  of 
the  number  is  quotient 

102.  1.  Divide  6374  by  1000. 

PROCESS.  Analysis. — Since  the  divisor  contains  no 

1l^^^^fil^7J.         ^rdcr  of  units  lower  tljan  thousands,  in  divid- 

' ing  we  may  omit  or  cut  od'  from  the  dividend 

"  To OIT  for  a  remainder,  all  orders  of   tinits   low^er 

than  thousands.     (Prin.) 
Dividing  6  thousands  by  1  thousand  we  obtain  6  for  the  quotient 
and  374  for  the  remainder,  or  G/^'g^. 

2.  Divide  39321  by  6000. 

^ ,  Analysis. — Since  the  divisor  con- 

PROCESS.  .                    t          ,         .       ,              , 

I A  A  A           I  o  tains  no  order  oi   units  lower  than 

6[0Q0)39|3^  1  thousands,  in  dividing  we  may  omit 

6      Rem.  3000  or  cut  ofF  from  the  dividend  all  or- 

3  21  ders  of  units  lower  than  thousands. 

'n  o  n  t  ^  thousands  are  contained   in   39 

Entire  remainder  33  21  ^,,^„,,„,j,   g   times   and    3    thousand 

remainder.    3  thousand  jdus  the  other  partial  remainder  equals  the 
entire  remainder.    Ileneo  the  quotient  is  C^JJ. 


70 


DIVISION. 


Rule. — Cut  off  the  eiphenfram  the  rigid  of  the  divisor,  and  as 
many  figures  from  the  right  of  the  dividend. 

Divide  the  rest  of  Uie  dividend  by  the  red  of  tiie  divisor. 

Annex  to  the  remainder  the  figures  cut  off;  the  result  will  he 
the  true  remainder. 


Divide  the  following: 

3.  Divide    1869  by  100. 

4.  Divide.  12345  by  200. 

5.  Divide  89325  by  700. 

6.  Divide  35968  by  900. 


7.  Divide    2465  by  1000. 

8.  Divide  13692  by  4000. 

9.  Divide  83005  by  1100. 
10.  Divide  75684  by  1500. 


11.  How  many  miles  of  railroad  at  $50000  a  mile  can  be 
constructed  for  $38968457? 

12.  How  many  schooners  carrying  8300  bushels  of  wheat 
will  it  require  to  carry  984364  bushels? 

13.  The  area  of  the  fetate  of  New  York  is  47UUU  t^iuare 
miles,  or  30080000  acres.     How  many  acres  in  a  square  mile? 

103.  Reijition  of  Dividend,  Divisor,  and  Quotient. 

The  value  of  the  quotient  depends  upon  that  of  the  divi- 
dend and  divisor.  If  one  of  these  is  changed,  while  the  other 
remains  the  same,  the  quotient  will  be  changed.  If  both  are 
changed,  the  quotient  may  not  be  changed. 

The  changes  may  be  illustrated  as  follows : 

FUNDAMENTAL  EQUATION. 

64-4-8  =  8. 


1.  Dividend 
changed. 


CHANGED  EQUATIONS. 


1.  128-^8  =  16] 


2.    32--8=   4 


1.  Multiplying  the  dividend 
by  2   multiplies  the  quotient 

!^  by  2. 

2.  Dividing  the  dividend  by 
2  divides  the  quotient  by  2. 


DIVISION 

71 

1. 

64   :    Ki         4 

1 .  Multiplying  the  divisor 

2.  DivUor 

by  2  divides  the  quotient  by 
2. 

2.  Dividing  the  divisor  by 

changed. 

2. 

64 -f-    4=16 

- 

- 

2  multiplies  tlio  qnotient  by  2. 

1. 

128 --16  =  8 

Multiplyinji:    or     dividing 

3.  Both 

both  dividend  and  divisor  by 

dianged. 

2  does  not  change  the  quo- 

2. 

32-^   4  =  8 

tient. 

From  these  illustrations  the  following  principles  are  de- 
duced : 

104r.  Prinx'IPLES. — 1.  Multiplying  the  dividend  or  dividing 
the  divisor y  multiplies  the  quotient. 

2.  Dividing  the  dividend  or  multiplying  Hie  divisor^  divides  Hie 
quotient. 

3.  MuUiplyhig  or  dividing  hotli  dividend  and  divisor  by  the 
same  number y  does  not  dmnge  the  quotient. 


ANALYSIS    AND   REVIEW. 

105.  Analysis  is  the  process  of  solving  problems  by 
tracing  the  relation  of  the  parts. 

In  analyzing  we  commonly  reason  from  the  given  number  to 
one,  and  then  from  one  to  the  required  number. 

1.  If  8  yards  of  cloth  cost  $16,  what  will  12  yards  cost? 

PROCESS.  Analysis.— Since  8  yards  cost  $16,  1  yard 

8  yards=S16.  will  cost  one-eighth  of  $1G,  or  $2;   and  since 

1       «'    ==8  2.  1  yard  costs  $2,  12  yards  will  cost  12  times 

12       '*    =824.  $2,*  or  $24. 

2.  If  8  horses  cost  ^2400,  what  will  6  horses  cost? 

3.  If  8  lemons  cost  40  cents,  what  will  11  lemon.s  cost? 

4.  How  much  will  12  hats  cost,  if  8  hats  cost  816? 


72  i>i\i.siuN. 

6.  If  25  pounds  of  sugar  cost  $2.50,  what  will  36  pounds 
cost? 

6.  If  12  men  can  build  a  school-house  in  25  day?,  how  lonq: 
will  it  take  25  men  to  build  it? 

*     7.  If  12barrol— **''-";rrtr- v--*»>  ^'  :-.  uha,  ,.,.    __  ..ai- 
rels  worth? 

8.  If  it  requires  57G  feet  of  boards  to  build  18  rods  of 
fence,  how  many  feet  will  be  required  to  build  13  rods? 

9.  If  6  men  can  do  a  piece  of  work  in  10  days,  how  long 
will  it  take  5  men  to  do  it? 

10.  If  I  exchanged  18  barrels  of  flour  for  61  yards  ol  cloili 
at  $i  a  yard,  how  much  did  I  get  per  barrel  for  the  flour? 

106.  The  Prtvrtithesis,  (  ),  shows  that  the  numbers 
included  within  it  are  to  be  subjected  to  the  same  operation. 

Thus,  ( o  -f  6  —  2 )  X  3  RhowH  that  5  +  6  —  2,  or  9,  Ib  to  be  multi- 
pi  ieil  by  3. 

107.  The  Vinculum f  ,  may  be  used  instead  of 
the  parenthesis. 

Thu8,  instead  oC(5  +  e  —  2)X3,we  may  write  64-6— 2X  3. 

Find  the  value  of  the  following: 

11.  (12-f  7  — 9)X5. 

12.  (13  — 6  +  8)  X  6. 

13.  (11  — 2-|-5)X8. 

14.  (3  +  4)x9  — (3  +  6)--3. 

15.  (54-7  — 3)x3  +  (3-{-5— 4)^4. 

16.  (36  — 7)X5  +  (102-f  6)-!-iT  

17.  (99  — 3)-^ 8  — (86  +  10)^12  ;ca-i- 6)  ^3. 

18.  (45  4-3)-^6-f  (10-hl5)-^(7  — 2)-f6. 

19.  A  man  <lying,  left  the  followinir  tracts  of  laud  to  be 
divided  equally  among  his  five  children.  The  first  tract  con- 
tained 1118  acres;  the  sei»ond,  3  times  as  much  lacking  193 


DIVISION.  73 

,  the  third,  twice  as  much  iis  the  other  two  lacking  105 
acres.     What  was  each  one's  share  ? 

20.  A  gentleman  bought  1516  head  of  cattle  at  331)  jkt 
head.  During  the  summer  07  died  of  disease,  but  he  sold  the 
renuiinder  so  as  to  gain  on  the  whole  number  $1819.  How 
much  did  he  get  for  his  cattle  per  head? 

21.  If  a  young  man  who  has  a  salary  of  $30  per  week, 
pays  S7.25  for  his  lM)ard  and  84.25  for  other  expenses,  how 
long  will  it  take  him  to  siivc  §1500? 

22.  A  man  bought  a  horse  for  $115  and  after  keeping  him 
three  months,  sold  him  for  3155.  If  he  paid  830  for  his  keep- 
ing and  received  850  for  the  use  of  him  during  that  time, 
how  much  did  he  gain? 

23.  A  s|)eculator  purchased  a  certain  number  of  bushels 
of  wheat  for  88735.  He  sold  it  for  89215  and  in  so  doing 
giiined  8.25  per  bushel.     How  many  bushels  did  he  buy? 

24.  I  bought  25  barrels  of  flour  for  8200.  For  what  must 
it  be  sold  per  barrel  to  gain  850?  What  will  be  the  gain  per 
barrel  ? 

25.  A  tailor  having  8585  wished  to  purchase  with  this  an 
equal  number  of  yards  of  two  kinds  of  broadcloth.  One  kind 
was  worth  86  a  yard,  the  other  87  a  ynnl.  How  nmuy  yards 
of  each  kind  could  he  buy? 

26.  Two  men  leave  the  same  place  at  the  same  time  and 
travel  in  opposite  directions,  one  at  the  rate  of  48  miles  per 
day,  the  other  at  the  rate  of  52  miles  per  day.  How  far 
a|)iirt  will  they  l)e  at  the  end  of  5  days? 

27.  If  20  men  can  do  a  piece  of  work  in  31  days,  how  many 
days  will  be  required  to  do  an  equal  amount  of  work  if  11 
additional  men  are  employed? 

28.  A  farmer  wished  to  obtain  8120.  He  sold  16  barrels  of 
apples  at  83.50  ymr  barrel,  and  enough  barley  at  8  .80  a 
bushel  to  make  up  the  sum  re(iuired.  How  many  l)ushels 
of  barley  did  he  sell? 


74  DIVISION. 

29.  Mr.  B.  bought  140  acres  of  land  for  $17500,  and  sold 
enough  at  8120  jxjr  acre  to  amount  to  89600.  The  rest  of  the 
land  he  sold  at  cost.  How  nuniv  5w*n>s  did  Iw  sill  nt  fost  ;n><l 
what  was  the  entire  loss? 

30.  A  man  pays  8628  a  year  lor  groceries,  So.jO  lor  house 
rent,  8262  for  clothes,  twice  as  much  for  traveling  expenses 
as  for  house  rent,  8175  for  annual  premium  for  life  insurance, 
and  saves  in  4  years  enougli  money  to  purchase  130  acres  of 
land  at  853  an  acre.     What  is  his  yearly  income? 

31.  In  October,  1871,  the  great  fire  in  Chicago  burned  ovet 
an  area  of  2124  acres.  The  estimated  loss  occasioned  by  the 
fire  was  8196000000.     What  was  the  average  loss  per  acre? 

\  boy  has  a  velocipede  which  he  can  run  at  the  rate 
ol  1  iu  rods  in  4  minutes.  How  many  minutes  will  it  take 
him  to  run  it  O^iO  rods? 

\  farmer  has  1000  head  of  cattle  in  5  fields.  In  the 
th<i  he  has  315  head,  in  the  second  175  head,  in  the  third 
300  head,  and  in  the  fourth  the  same  number  as  in  the  fifth. 
How  many  has  he  in  the  fifth  ? 

34.  A  man  gave  away  845000  in  three  efpial  amounts. 
One  share  he  gave  to  his  son,  one  share  to  his  daugliter,  and 
the  rest  to  his  grandchildren,  giving  them  81500  apiece. 
How  many  grandchildren  had  he? 

35.  In  the  Centennial  Exhibition,  at  Philadelphia,  a  section 
of  a  cable  in  process  of  construction  for  the  new  suspension 
bridge  at  New  York  was  shown.  It  was  composed  of  6000 
galvanized  steel  wires,  and  its  ultimate  strength  was  22,300,000 
pounds.     What  weight  was  each  wire  capable  of  sustaining  ? 

36.  The  main  building  of  the  Centennial  Exhibition  at 
Philadelphia,  the  largest  building  in  the  world,  contained  on  the 
ground  floor  an  area  of  872320  square  feet,  on  the  upper  floors 
in  j)rojections  37344  s(|uare  feet,  in  towers  20344  square  feet. 
If  there  are  43560  square  feet  in  an  acre,  how  many  acres  did 
the  floors  of  the  building  contain  ? 


\ 


PROPERTIES  OF  NUMBERS 


"^/ 


'/\" 


108.    1.  Wliat  is  the  product  of  4  times  5?    What  are  4 
and  5  of  their  product? 

2.  What  is  4  of  16?    Of  24?    What  is  7  of  14?    Of  28? 

3.  What  numbers  will  exactly  divide  18?     24?     36?     72? 

4.  Give  the  exact  divisors  of  42.    96.     108.    48.    32?. 

5.  What  are  the  factors  of  30?    24?     40?    56?     64? 

6.  What  numbers  between  0  and  10  can  not  be  divided  by 
any  numlxir  except  themselves  and  1?     Between  10  and  20? 

7.  What  numlx^rs  between  0  and   10  can  be  divided  by 
other  numbers  than  themselves  and  1  ?     Between  10  and  20? 


.  DEFINITIONS. 

109.  An  Integer  or  Integral  Number  is  one  that 
expresses  whole  units. 

Thus,  281,  36  houses,  4G  men,  are  integral  numbers. 

110.  An  Eoract  Divisor  of  a  number  is  an  integer  that 
will  divide  it  without  a  remainder. 

Thus,  2,  4,  G  and  12  are  exact  divisors  of  24. 

111.  The  Factors  of  a  number  are  the  integers  which 
being  multiplied  together  will  produce  the  number. 

Thus,  6  and  8  are  factors  of  48. 


The  exa^t  divisors  of  a  number  are  factors  of  it. 


(75) 


76  PUOPERTI1C8  OF   NUMBERS. 

112.  A   Prhne  Namhev   is  one  that  has  no  exact 
divisors  except  itself  and   1 

Thus,  1,  3,  5  and  7  are  prime  nuiuinn*. 

113.  A  Composite  Number  i?  one  that  has  exact 
divisors  besides  itself  and  1. 

Thus,  18  and  24  are  coniiioeitc  numbers,  for  18  b  divisible  by  6,  and 
24  by  8. 

114.  An   /wry/    A^/yi//>rr  is  one  that  is  exactly  divisible 
by  2. 

Thus,  2,   1.  '■..  -.  .  •  :i  nimiKr.^. 

115.  An  Odd  Number  is  one  that  is  not  exactly  divis- 
ible by  2. 

ThuH,  1,  3,  5,  7,  9,  etc^  are  odd  numbers. 


DIVISIBILITY  OF  NUMBERS. 

116.  In  determining  by  inspection  the  divisibility  of  num- 
Irms,  the  following^facts  will  be  found  valuable. 

1.  Two  is  an  exact  divisor  of  any  even  number. 
Thus,  2  is  an  exact  divisor  of  12, 16,  30  and  44. 

2.  Three  is  an  exact  divisor  of  any  number,  the  sum  of  whose 
digits  is  divisible  by  3. 

Thus,  3  is  an  exact  divisor  of  312,  135,  423,  and  3816. 

3.  Four  ']-  an  exact  divisor  of  a  number,  if  the  number 
expressed  by  its  two  right  hand  figures  is  divisible  by  4. 

Thus,  4  i8  an  exact  divisor  of  264,  1284,  1368,  and  7932, 

4.  Five  is  an  exact  divisor  of  any  niinihcr  whose  riirlit  hand 
figure  is  0  or  5. 

Thus,  5  is  an  exact  divisor  of  360,  1795,  3810,  and  7895. 


I'lVifiiiiiLl  1  ^'    "|-    M   Mi'.i:i:s.  il 

5.  tSix  is  ail  exact  divisor  of  any  even  ihiihIkt,  the  sum  of 
whose  digits  is  divisible  by  3. 

Thus,  0  ifl  an  exact  divisor  of  732,  534,  798,  and  8226. 

6.  Eight  is  an  exact  divisor  of  a  number,  if  the  number 
expressed  by  its  three  right  hand  figures  is  divisible  by  8. 

Thus,  8  is  an  exact  divisor  of  4328,  3856,  61360,  and  5920. 

7.  Nine  is  an  exact  divisor  of  any  number,  the  sum  of 
whose  digits  is  divisible  by  9. 

Thus,  9  is  an  exact  divisor  of  513,  1314,  252,  1341,  ami  312462. 

8.  10,  100,  1000,  etc.,  are  exact  divisors  of  any  numbers 
that  end  respectively  with  one,  two,  three,  etc.,  ciphers. 

Thus,  10,  100,  1000,  etc.,  are  exact  divisors  respectively  of  80,  800, 
8000,  etc. 

9.  If  an  even  number  is  divisible  by  an  odd  number  it  is 
divisible  by  twice  thai  number. 

Thus,  72  is  divisible  by  9  and  by  twice  9  or  18.     312  by  3  and  6. 

10.  An  exact  divisor  of  a  number  is  an  exact  divisor  of  any 
number  of  times  that  number. 

Thus,  3  is  nn  o^cact  divisor  of  12,  and  of  any  numhii  m   imM  ~  !_'. 

EH  36. 

11.  An  exact  divisor  of  each  of  two  numbers  is  an  exact 
divisor  of  their  mm  and  of  their  difference. 

Thus,  3  Is  an  exact  divisor  of  9  and  12  respectively,  and  therefore 
<>t  ••       IJ   uv  21;  of  12  —  9,  or  3. 

117.  Find  by  inspection  some  of  the  exact  divisors  of  the 
following  nuTnlxis: 


1.  1524. 

..r)6. 

9.  42840. 

13.  376250. 

2.  3432. 

6.  7236. 

10.  92475. 

14.  428328. 

3.  4264. 

7.  27360. 

11.  362088. 

15.  4183200. 

4.  9360. 

8.  23661. 

12.  438408. 

16.  6853744. 

78  rROPEUTIES   OF   NUMBERS. 


FACTOlil.NG. 

118.  1.  What  are  the  factors  of  6?     8?     12?     16? 

2.  What  factors  of  18  are  prime  numbers  or  primo  fnctor^? 

3.  Wliat  are  the  prime  factors  of  30? 

4.  What  are  all  the  exact  divisors  of  30? 

5.  What  numbers  besides  the  prime  factors  of  30  are  its 
exact  divisors?  How  are  they  obtained  from  the  prime 
factors? 

6.  Of  what  number  are  2,  3,  and  .">,  ihc  piiuic  iiictors? 

7.  How  can  a  number  he  obtained  from  its  prime  factors? 

8.  The  prime  factors  of  a  number  are  2,  2,  and  5.  What 
is  the  numlx^r?    Give  all  the  exact  divisors  of  this  number. 

9.  What  are  the  exact  divisors  of  60?    72?    96?     144? 

DEFINITIONS. 

119.  Factor  hi  g  is  the  process  of  separating  a  number 
into  its  factors. 

1*20.  Prime  Ixictors  arc  llutors  that  are  prime 
nuinlxTs. 

1*^1.  The  number  of  times  a  number  is  used  as  a  factor  is 
indicated  by  a  small  figure  called  an  exponent.  It  is  written 
above  and  at  the  right  of  the  numl>< 

Thus,  4X4X4=4'\  and  the  3  imli.:;.  ,.  r.  i  i~  n  r.l  a>  n  factor 
three  tiiiiei^. 

122.  Principixs. — 1.  Every  prime  fador  of  a  nnmher  is  an 
exojct  divisor  of  tJiat  number. 

2.  The  only  ejcad  divisors  of  a  nwv^  ■  ixriors  or 
the  product  of  two  or  more  of  them. 

3.  Every  number  is  equal  to  the  product  of  its  prime  factors. 


FACrORINCi. 


79 


1.  What  are  the  prime  factors  of  756? 

Analysis. — Since  every  prime  factor  of  a  number 
is  an  exact  divisor  of  the  number,  we  may  find  the 
prime  factors  of  75G  by  finding  all  the  prime  numbers 
that  are  exact  divisors  of  75G.  Since  the  number  is 
eren,  we  divide^  by  2.  Since  the  quotient  obtained  is 
an  even  number,  we  divide  again  by  2.  Then  we  di- 
vide by  the  prime  numbers  3,  3,  3,  successively,  and 
the  last  quotient  is  7,  which  is  a  prime  number. 

Hence  the  prime  factors  are  2,  2, 3, 3, 3, 7,  or  2%  3^,  7. 


PROCESS. 

2)756 

2)378 

3)189 

3)63 

3)21 

7 


Rule. — Divide  Vie  given  number  by  any  prime  number  that 
will  exxictly  divide  it.  Divide  Hm  quotient  by  a)wther  prime  ««m- 
ber,  and  so  continue  until  the  quotient  is  a  prime  number. 

The  several  divisors  and  last  quotient  will  be  the  prime  factors. 


What  are  the 


2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 


Of  35? 
Of  64? 
Of  336? 
Of  168? 
Of  144? 
Of  315? 
Of  198? 
Of  224? 
Of  786? 
Of  316? 


prime  factors 

12.  Of    484? 

13.  Of  1280? 

14.  Of  1008? 

15.  Of  1140? 

16.  Of  1184? 

17.  Of  1872? 

18.  Of  7644? 

19.  Of  2310? 

20.  Of  3204? 

21.  Of  4725? 


22.  Of 

23.  Of 

24.  Of 

25.  Of 

26.  Of 

27.  Of 

28.  Of 

29.  Of 

30.  Of 

31.  Of 


3913? 

3812? 

7007? 

3980? 
26840? 
38148? 
11340? 
24024? 
18500? 
124416? 


MTTTTTPTJrATTOV  T^Y  FACTORS. 


Vl',\,    1.   Wl.at  arc  iiu-  laciors  ..i  lit     16?     18?     20? 

2.  What  are  the  factors  of  24?     42?     36?    30?    27? 

3.  What  are  the  factors  of  45?     48?    56?     63?     72? 

4.  When  a  number  is  multiplied  by  4  and  tlio  ]>roduct  by 
6,  by  what  is  the  number  multiplied? 


80  PROPKRI  NITMBKRS. 

6.  What  will  20  carriages  cost  at  $346  each? 
PROCESS.  Analysis.— >Since  20  ir  5  timm  4,  20 

$346  coct of  1  carrinsc.       carriago«  will  coHt  5  linicH  ha  much  a.-* 
4  4  carriagw.     4  carriagis   will   cost  4 

$1384  ooBtoficaniaees.      ^-n  ,^,  r !.  ,        ,         . 

-  will  eo«t.o  times  as  much  as  4  carria^^cn. 

or    5   times    $i:J8l,    which   is   $6920. 

$6920  cottofaocarrlagc*.     Hence,  20  imiiLrs  will  n.st  $6920. 

RuLK. — Multiply  Uie  multiplicaml     ,  the  mtdtU 

jilier,  (lie  product  diwi  obtain^  by  aiioUier  factory  and  so  continue 
until  aH  the  factors  have  been         '  J  rely  as  mtdtipliers. 

The  lai*  prodkuA  will  he  tJie  prod 

Multiply  in  same  manner,  using  the  factors  of  the  multiplier: 

6.  425  hy  32;  by  36;  by  48;  by  72. 

7.  1S24  by  56;  by  27;  by  I  .;  I  y  108. 

8.  What  will  be  the  cost  of  35  cows  at  $64  each? 

9.  What  will  21  cortls  of  wood  cost  at  85.35  a  cord? 

10.  What  will  72  yoke  of  oxen  cost  at  $168  per  yoke? 

11.  What  will  36  boxes  of  lemons  cost  at  $6.25  \^r  box? 

12.  What  will  48  acres  of  land  cost  at  $46  per  acn  ? 

13.  What  will  24  jMiintings  cost  at  $55  each? 

14.  What  will  45  cases  of  boots  cost  at  $36  a  case? 

15.  What  will  56  barrels  of  salt  cost  at  $2.35  a  barrel? 


DIVISIOX  BY  FACTORS. 

124.   1.  What  are  the  factors  of  32?    25?    64?    96? 

2.  If  a  number  is  divided  by  8,  by  what  must  the  quotient 
be  divided  that  the  number  may  be  divided  by  16? 

3.  If  a  number  is  divided  by  8  and  the  quotient  by  6,  by 
H'hat  is  the  numl>er  divided? 

4.  What  factors  may  be  used  to  divide  a  luiinber  by  30  ? 

5.  What  fectors  may  be  used  to  divide  a  number  by  48  ? 


•■ ';iN<;.  81 

6.  A  milh-r  ]tiii  iiji  .)*/.f  pniiii(i.s  (,!'  lioininy  in  packages  con- 
taining 4  jxnimls  each,  and  iwieked  tlieni  in  l)oxes  containing 
10  packages  eacli.  How  many  packages  and  how  many 
Ih)xcs  did  he  liave  ? 

7    Divide  888  by  24,  using  factors. 

PROCESS.  Analysis. — 24   is  equal   to  6  times  4.    JTenco  to 

4)888  divide  by  24  we  may  divide  by  6  times  4.    888  -f-  4  = 

.  — ——  222.     But  Bince  we  were  to  divide   by  6  times  4,  this 

I quotient  is  6  times  too  great,  hence  we  nmst  divide  it  by 

3  7  6.    222  -^  6  =  37  the  true  quotient. 

8.  Divide  5863  by  32,  using  factors. 

PROCESS.  Analysts. — 32  is  equal  to  4  X 

4)5683  2X4.     Dividing  5083  by  4  gives 

.                       ■  a  quotient  of    1420  foum  and  3 

^  LLz^^  "  '^  units  remaining. 

4)710  Dividing  1420  fours  by  2  gives 

i~77         2  ^  quotient  ot  710 etyhlii.    Dividing 

3  +  (  2  X  8  )  =  1  9  true  Rom.     ™  'i^f"' '7,  *  five's  a  quoli^t 

'    ^  ^  of     1//    thirtif-tuox   and    2   ciy/it-i 

1  7  7  if  Quotient,      remainder.     The  first  partial  re- 

mainder  is  3  units,  and  the  second,  2  jcUjhtA,  or  16  ;  hence  the  entire 

rcjnainder  is  3  -|-  16,  or  19,  and  the  quotient  is  177|^. 

Rule. — Divide  the  dividend  by  one  factor  of  the  diviso)',  ihe 
quotient  thm  obtained  by  anxfthei'  factor j  and  so  continue  until  all 
the  factors  have  been  nsed  succemvehj  as  divii^ors. 

If  there  be  remainders,  multiply  each  remainder  by  all  the  preced- 
inrj  divisors  except  the  one  that  produced  it.  The  sum  of  Viese  prml- 
ucts  will  be  tJie  true  remainder. 


Divide,  using  factors: 

\).   1704  by  24. 

13. 

1288  by  56. 

17. 

3275  by  56. 

0.  4725  bv  15. 

14, 

3528  by  72. 

18. 

3276  by  27. 

1.  5740  by  28. 

15. 

3824  by  32. 

19. 

4104  bv  45. 

2.  1428  by  42. 
G 

16. 

2184  by  49. 

20. 

7304  by  24. 

82  PROPERTU:  MBERS. 

21.  A  wholesal  put  up  1120  pounds  of  tea  in 
35-poun(l  jxickagcjj,  cuiiiaining  t^-iwund  canister-.  TI-w 
many  jwckagcs  and  canisters  were  there? 

22.  A  paper  manufacturer  put  up  his  paper  so  that  (ah 
quire  contained  4  packages  of  6  sheets  each.  How  many 
packages  and  quires  were  made  up  from  912  sheets? 


CANCELLATION. 

125.  1.     liuA     liiiiliV     liiii*  .      i.M.v-    .>    ..M,uiim*l     in    4 

times  5?  2  times  7  in  4  tin  7  2  times  9  in  4  times  9? 
2  times  24  in  4  times  i'  iiues  auy  number  in  4  times 

that  number? 

2.  How  many  times  i-    \    times  8  contained  in  I'J 
8?     4  times  25  in  12  times  25?     4  times  75  in  12  times  75? 
4  times  any  number  in  12  times  the  same  number? 

;    How  many  times  is  6  X  48    contained    in    24x48? 
6  X  144  in  24  X  144? 

4.  In  determining  the  quotient,   what  uin'^'  »-   "''v  1)e 
omitted  from  both  dividend  and  divisor? 

126.  Cancellation  is  the  process  of  shortening  compu- 
tations by  rejecting  equal  factors  from  the  dividend  and  divisor. 

127.  Principle. — Rejecting  equal  factors  from  both  dividend 
and  divisor  does  not  alter  Uie  value  of  the  quotient. 

1.  TVividc  66  times  36  by  24  times  11. 

PBOCESS.  Analysis.— We   write 

66X36       1 1  X  ^  X  ^  X  3  X  3  1^^  numbers  a.s  in  divis- 

~        TT      ■  ~ .  "2 Z~r  —  9     ion,  the  dividend  above, 

'      A      7*      A      /*  A  jj^g  divisor  below  a  line, 

instead  of  multiplying  06  by  36  we  resolve  60  into  its  factors  1 1  and 
6,  and  36  into  its  factors  4,  3  and  3,  and  in  the  divisor  resolve  24  into 
the  factors  6  and  4. 


CAM.hl.L.\lH;>.  83 

Cancelling  equal  factors  from  both  dividend  and  divisor,  which  in 
the  same  as  dividing  both  by  the  same  number,  and  does  not  alter 
the  value  of  the  quotient,  we  have  remaining  in  the  dividend  the 
factors  3  and  3,  or  9,  which  is  the  quotient. 

2.  Divide  72  X  66  X  49  by  63  X  40  X  21. 

PROCESS.  Analysis. — We  write  the  num- 

g         22        7  ^^  ^^  before.    Since  9  is  a  factor 

7'2vft^v4<?       22  ^^  ^^^  "^  *"^  ^^  ^'  ™^^  ^  rejected 

;f  ^  App  A^^  ^  —  ^^      ^^^^  ^^^^  leaving  8  instead  of  72  in 

^^X^PX?^         5  the  dividend,  and  7  instead  of  63  in 

/T  5  ^  the  divisor.     We  next  cancel  8  from 

8  and  40,  leaving  5  instead  of  40  in  the  divisor.  We  next  cancel  7 
from  7  and  49,  leaving  7  instead  of  49  in  the  dividend,  and  7  again 
from  7  and  21,  leaving  3  instead  of  21.  Rejecting  the  factor  3  from 
both  G6  and  3,  there  is  left  for  a  dividend  22,  and  for  a  divisor  5,  which 
gives  a  quotient  of  4 J. 

Rule. — Reject  from  die  dividend  and  divisor  aU  factors  common 
to  both,  and  then  divide  ilie  product  of  the  remaining  factors  of  Vie 
dividend  by  the  product  oftJie  remaining  factors  of  the  divisor. 

When  all  the  factors  of  both  dividend  and  divisor  are  cancelled,  the 
quotient  is  1,  for  the  dividend  will  then  exactly  contain  the  divisor 
once. 

EXAMPLEa 

Divide,  using  cancellation : 

3.  7X5X3X11  by  5X11X3. 

4.  12  X  14  by  6  X  7  X  2. 

5.  6  X  3  X  5  X  2  by  3  X  5  X  2  X  2. 

6.  4  X  2  X  8  X  24  by  36  X  8  X  2. 

7.  24  X  32  by  8  X  6  X  4. 

8.  45  X  60  X  7  by  49  X  12  X  9. 

9.  2X3X5X8X7  by  6X5X2X7. 

10.  5  X  8  X  12  X  6  by  20  X  16  X  2. 

11.  12  X  60  X  36  X  35  by  7  X  30  X  18  X  24. 

12.  30  X  49  X  64  X  25  by  35  X  15  X  24. 


84  PROPERTH>  MBERS. 

13.  Divide  the  product  of  26  times  18  times  35,  by  78 
times  30. 

14.'  Find  the  quuiuuL  m  .♦;*  times  360  times  365,  divided 
by  11  times  72. 

15.  Find  the  quotient  of  175  X  28  X  72  times  363,  divided 
by  12  X  11  X  9. 

16.  Four  ikrms  containing  80  acres  each,  worth  $65  per  acre, 
were  exchanged  for  5  farms  containing  95  acres  each.  What 
was  the  value  per  acre  of  the  farms  received  in  exchange? 

17.  A  farmer  buys  3  pieces  of  muslin  each  containing  44 
yards  at  11  cents  a  yard,  and  pays  for  it  in  wheat  at  $2  per 
bushel.     How  many  bushels  are  recjuired? 

18.  A  merchant  bought  13  tubs  of  butter,  each  containing 
89  i>ound!?,  at  32  cents  a  pound,  jwying  for  it  in  4  imtterns  of 
silk  of  13  yards  each.     How  nnu  li  wn-  ♦'-    •'!:  a  van  I? 


ri»\nfn\  dfvtsorr. 

128.     1.  What  nuniUrs  will  exactly  divide  12?     15?     20? 

2.  Wliat  numlicrs  will  exactly  divide  both  12  and  1")?    15 
and  20?    24  and  48?    63  and  72? 

3.  What  numlxjrs  will  exactly  divide  both    12   and   24? 
Wliat  is  the  largest  number  that  will  exactly  divide  them? 

4.  What  is  the  largest  number  that  will  exactly  divide 
both  15  and  30?     16  and  32?     16  and  24?     24  and  32? 

5.  Name  all  the  divisors  common  to  15  and  30. 

6.  Name  all  the  prime  divisors  or  factors  common  to  15 
and  30? 

7.  How  is  the  greatest  divisor  common  to  15  and  30  found 
from  the  prime  factors  of  those  numbers? 

8.  AVhat  is  the  greatest  divisor  common  to  24  and  30? 

9.  How   is   the  greatest  divisor  common   to   24  and   30 
obtained  from  the  prime  factors  of  those  numbers? 


COMMON    J)iVlr>oli.^.  85 

DEFINITIONS. 

129.  A  Common  Divisor  of  two  or  more  numbers 
is  an  exact  divisor  of  each  of  them. 

Thu«,  G  is  a  common  divisor  of  12,  24,  48  ;  8  of  10,  24  and  64. 

130.  The  Greatest  Common  Divisor  of  two  or 

more  numbers  is  the  greatest  number  that  is  an  exact  divisor 
of  each  of  them. 

ThuH,  24  is  the  greatest  common  diviHor  of  24  and  48. 

131.  When  numbers  have  no  common  divisor  they  are  said 
to  be  I*rime  to  each  other. 

Thus,  7,  8  and  9  are  prime  to  each  other. 

A  common  divisor  is  sometimes  called  a  common  measure  and 
the  greatest  common  divisor  the  greatest  common  measure. 

132.  Principle. — The  greatest  comvwn  divisor  of  two  or  more 
numbers  is  tlie  p'oduct  of  all  their  common  prime  factors. 

1.  AVhat  is  the  greatest  common  divisor  of  45,  60,  and  75? 

1st  process.  Analysis. — Since  the  greatest  com- 

45__3\/3v/5  mon  divisor  is  equal  to  the  product  of 

/>Q__OvxOv^V^    *^^  ^^^   prime  factors  common  to  the 

^ ft       K  V?  K  given  numbers,  we  separate  the  num- 

A  y  A  i*  ijers  into  their  prime  factors.    The  only 

"     '  5  =  15  prime  factors  common  to  all  these  num- 

bers arc  3  and  5.     Hence  their  product, 
15,  ia  the  greatest  common  divisor  of  the  given  numbers. 

2d  process.  Analysis. — 3  will  divide  each  of  the 

45       60       7   '  iven  numbers,  and  is  therefore  a  factor 

of  the  greatest  common  divisor.    5  will 


divide  each  of    the  resulting  quotients 


•>  4  5  and  is  therefore  a  factor  of  the  greatest 

,'}  X  5  =  15  common    divisor.     The  quotients   3,   4, 

and  5,  have  no  common  divisor;  there- 
fore 3  and  5  are  the  only  factors  of  the  greatest  common  divisor,  15. 


86 


PRUPERTIE- 


Rule. — Separate  the  numbers  hUo  tiieir  prime  factors  and  find 
the  product  of  all  Hie  common  fadors,     Or^ 

Divide  Ute  numbers  by  any  common  divisory  die  resulting  quo- 
tients by  aivoUier  common  divisor,  and  so  continue  to  diviile  until 
quotients  are  obtained  that  have  no  comnwn  divisor. 

Tfieprodtui  of  the  divisors  will  be  the  greatest  common  divisor. 


EXAMPLES. 
What  is  the  greatest  common  divisor  of 


2.  12,  16,  20 

3.  18,  27,  36,  -io 

4.  24,  48,  60,  72/ 

5.  36,  60,  72,  66? 

6.  48,  72,  96,  84? 

7.  18,  81,  72,  54? 

8.  32,  48,  80,  96? 

9.  45,  63,  99,  81? 
10.  35,  56,  84,  63? 


11.  HI,  40, 

12.  3t5,  60, 

13.  30,  55, 

14.  80,  54, 

15.  14,  42, 

16.  24,  28, 

17.  33,  77, 

18.  24,  72, 

19.  42,  84, 


72,    88? 

84,  96? 

85,  90? 
66,  78? 
63,    01? 

120,  144? 
143,  154? 
120,  168? 
252,  204? 


133.  When  the  numbers  cnn  not  W  f-i 
following  method  is  employed : 


I'lilv,  the 


1.   ^Vhat  is  the  greatest  common  divisor  of  35  and  168? 

PBOCBSS. 


85)168(4 
140 


28)35(1 
28 


Analysis. — The  greatest  common 
divisor  can  not  be  greater  than  the 
Bmaller  number;  therefore  35  will  be 
the  greatest  common  divisor  if  it  i» 
exactly  contained  in  168.  By  trial  it 
is  found  that  it  is  not  an  exact  divisor 
of  168,  since  there  is  a  remainder  of 
28.  Therefore  35  is  not  the  greatest 
common  divisor. 

Since  168  and  140,  which  is  4  times  35,  are  each  divisible  by  the 
greatest  common  divisor,  their  difTcrencc,  28,  must  contain  the  greatest 
common  divisor;  therefore  the  greatest  common  divisor  can   not  be 


7)28(4 
28 


COMMON  DIVISORS. 


87 


^,.v.... . _  .    .:>  will  be  the  greatest  common  divi.x.i  if  it  is  exactly 

cohtainwl  in  IWi;  Bince  if  it  be  contained  in  35,  it  will  be  contained  in 
140,  and  in  28  pliut  140,  or  1G8.  By  trial  we  find  that  it  is  not  an  exact 
divisor  of  J55,  for  there  is  a  remainder  of  7.  Therefore  28  is  not  the 
greatest  common  divisor. 

Since  28  and  .'io  are  each  divisible  by  the  greatest  common  divifioi-, 
their  diflercnce,  V,  must  contain  the  greatest  common  divisor;  therefore 
the  greatest  common  divisor  can  not  be  greater  than  7.  7  will  be  the 
greatest  common  divisor  if  it  is  exactly  contained  in  28;  since  if  it  be 
contained  in  itself  and  28,  it  will  be  contained  in  their  sum,  35,  and  also 
in  108,  which  is  the  sum  of  28  and  4  times  35,  or  140.  By  trial  we  find 
that  it  is  an  exact  divisor  of  28.    Hence  7  is  the  greatest  common  divisor. 

Rule. — Divide  the  greater  number  by  (lie  less  and  if  tliere  be  a 
remainder  divide  the  less  number  by  it,  then  Hie  preceding  divisw 
by  tlie  last  remainder,  and  so  on,  till  notliing  rnnain.'*.  The  last 
divisor  ^vill  be  tlie  greatest  common  divisor. 

If  more  than  tivo  nwnbers  are  given,  find  the  greatest  common 
divisor  of  any  two,  Uien  of  this  divisor  and  another  of  the  given  num- 
bers, and  so  on.    The  last  divisor  will  be  Uie  greatest  common  divisor. 


Find  the  greatest  common  divisor  of 


169  and  195. 
187  and  209. 
372  and  492. 
119  and  187. 
243  and  297. 
322  and  391. 


•8. 
9. 
10. 
11. 
12. 
13. 


252  and  294. 

156  and  208. 

702  and  945. 
1029  and  1197. 
1666  and  1938. 
3596  and  3768. 


What  is  the  greatest  common  divisor  of 


14.  672,  352,    992? 

15.  714,  867,  1088? 

16.  462,  759,  1155? 


17.  630,  1134,  1386? 

18.  462,  1764,  2562? 

19.  7955,  8769,  6401? 


20.  In  a  village  some  of  the  walks  are  56  inches  wide, 
some  70  inches,  and  others  84  inches.  What  is  the  width  of 
the  widest  flagging  that  will  suit  all  the  walks? 


88 


PKOPKUTIES   OF   NUMBERS. 


-1.  A  iiK  i(  li;iiii  has  60  pounds  of  tea  of  one  kind.  7  > 
jwunds  of  another,  and  100  pounds  of  another,  which  he 
wishes  to  put  up  in  the  largest  jx)S:5ible  equal  jmckagcs  with- 
out mixing  the  different  kinds.  How  many  pounds  should  be 
j)ut  in  each  jiackage? 

22.  Mr.  A.  has  324  acres  of  land  in  one  farm  and  7^  ul  i  l.-:  in 
another.  He  wishes  to  divide  these  into  the  largest  possible 
fields  of  equal  size.  How  many  fields  will  there  be,  and  how 
niiinv  acrt's  in  each  field? 


MULTIPLES. 

134,  1.  What  numlx^rs  less  than  25  will  exactly  contain 
4?    5?    6? 

2.   What  numbers  less  than 
nndfi? 

line  some   numbers  that  a 
Vv  4.     liy  both  5  and  4. 

4.  Name  some  numl)ers  that  : 
By  3.     By  4.     By  2  and  ; 

5.  What  is  the  smallest  immrKr  tnat  i- 
each  of  the  numbers  2,  3,  and  4  ? 

6.  What  is  the  least  number  that  will  contain  10  and  15? 

7.  \Vhat  common  prime  factors  have  10  and  15?  What 
factor  occurs  in  10  that  does  not  in  15?  What  factor  is  found 
in  15  that  is  not  found  in  10? 

8.  AVhat  are  all  the  different  prime  factors  of  10  an«l  !.'>/ 

9.  How  may  the  least  number  that  will  contain  10  and  15 
be  formed  from  their  prime  factors? 

What  is  the  least  number  that  will  exactly  contain 


tiy  contain  hdh  4 

ily  divisible  by  5. 

ily  divisible  by  2. 

exactly  (livif^ible  by 


10.  3,  6  and  9? 

11.  3,  5  and  6? 

12.  4,  8  and  12? 


13.  2,  3,  5  and  6? 

14.  3,  4,  5  and  6? 

15.  3,  6,  8  and  12? 


MULTIPLES.  89 


DEFINITIONS. 

1»35.  A  Multiple  of  a  number  is  a  number  that  will 
exactly  contain  it. 

A  multiple  of  a  number  is  obtained  by  multiplying  the  given  number 
by  some  integer. 

136.  A  Common  3Iultiple  of  two  or  more  numbers, 
is  a  number  that  will  exactly  contain  each  of  them. 

137.  The   Least  Common  Multiple  of  two  or 

more  numbers,  is  the  least  number  that  will  exactly  contain 
each  of  them. 

138.  Principle. — The  lead  common  multiple  of  two  or  viore 
numbers  is  equal  to  the  product  of  all  tlie  prime  factors  of  the 
ntunberSj  aiul  no  otlier  factors. 

WJilTTrX    rXJEltCISES. 

139.  1.  Find  the  least  common  multiple  of  30,  28  and  60? 

1st  process.  Analysis. — Since  the  least  cora- 

-     30  =  2X3X5  "^^^  multiple  is  equal  to  the  product 

Q  ft 9  N/  9  s/  7  of  all  the  different  prime  factors  of  . 

ooo       r:  ^^^   numbers  and  no  other  factors, 

60  =  2X2X3X5  (Prin.)  the  numlwrs  must  be  scpa- 

2X2X3X5X7  =  420      rated   into  their  prime  factors,  and 

the  product  of  all  the  different  prime 
factors  found.  The  prime  factors  of  GO,  the  largest  number,  are  2,  2, 
3  and  5.  28  contains  a  factor,  7,  which  is  not  found  in  60.  60  con- 
tains all  the  factors  of  the  other  numl)er,  30.  Therefore  all  the 
different  prime  factors  of  the  given  numbers  are  2,  2,  3,  5  and  7, 
and  their  product,  420,  is  the  least  common  multiple. 

Rule. — Separate  the  rjiven  numbers  into  their  prime  factors. 

Find  tJie  product  of  all  die  different  prime  factors,  using  eadi 
factor  the  greatest  number  of  times  it  occurs  in  any  of  thr  rprm 
numbers. 


90 


PROPERTIES   OF    NUMBEIJS. 


Find  the  least  common  multiple  of 

2.  28,  32  and  64.  r,.  12,  IG,  18  and  24. 

3.  36,  72  and  14  I  •.  15,  20,  25  and  30. 

4.  45,  70  and  9n  7  1«,  54,  90  and  180. 


What  is  the  least  roniinon  iiiuaijin' 


8. 

22,  55  and  77? 

13. 

12,  16,  24  iiUii  .M, . 

9. 

48,  60  and  180? 

14. 

32,  oGy  64  and  80? 

10. 

10,  64  and  96? 

15. 

14,  35,  50  and  28? 

11. 

81,  63  and  135? 

16. 

33,  99,  84  and  135? 

12. 

25,  70  and  95? 

17. 

17,  51,  65  and  121? 

18.  Find  the  least  common  niultiii 
2d  proces'S. 


.;o. 


2 

16     20 

2 

8     10      1 

5 

4        5     1 

>N 

4        1        G 
'5X4X3  =  240 

Analysis. — Since  2  is  a  prime  fac- 
>r  of  each  of  the  numbers,  it  is  also  a 
utor  of  the  lea.Ht  common  multiple. 
I'rin.)     Dividing,  there  remain   as 
the  other  factors  of  the  numbers,  8, 
10,  and  15.    2  is  a  prime  factor  of  8 
and  10,  and  is  therefore  a  factor  of 
the  least  common  multiple.    Divid- 
ing, there  remain  4, 5,  and  15. 
',  ,^  .i  i-iiiue  factor  of  .">  .nul  lo,  and  is  therefore  another  factor  of 
the  least  common  multiple.    Dividing,  there  remain  4,  1,  and  3,  \rhich 
are  prime  to  each  other.    Therefore  the  product  of  the  factors  2,  2,  5, 
4,  and  3,  will  be  the  least  common  multiple. 

Rule. —  Wriie  the  given  numbers  in  a  horizontal  line.  Divide 
by  any  prime  number  that  is  an  exact  divisor  of  tux)  or  more  of  the 
given  numbers^  andwn*    ''  -  ...  /.v..».   .  .  /  ;;..,•;../  ,.,..,,/,,,..  ;,j 

a  lin3  beneath. 

Thus  continue  to  divide  until  Hie  quotients  and  undivided  num- 
bers are  prime  to  eacJi  otJier.  The  product  of  Vie  divisors,  and 
the  numbers  in  Hie  last  fiorizontal  line^  will  be  tJie  least  commxm 
multiple. 


MLLTIPLES.  91 

In  finding  tlu*  least  common  multiple,  all  numbers  that  are  factors 
of  other  given  numbers  may  be  disregarded.  Thus,  the  multiples  of 
8, 16,  32,  64,  80,  and  128  arc  the  same  as  the  multiples  of  80  and  128. 


EXAMPLES. 


Find  the  least  common  multiple  of 


19.  60,  40,  120  and  72. 

20.  81,  45,  108  and  135. 

21.  40,  60,    80  and  120. 


22.  32,  36,  72  and  80. 

23.  30,  75,  60  and  90. 

24.  24,  44,  65  and  100. 


What  is  the  least  common  multiple  of 


j:..     8, 12, 16,  24  and  48? 
26.  16,  20,  24,  32  and  40? 


27.  25,  40,  75,  80  and  120? 

28.  32,  45,  70,  64  and    90? 


29.  What  is  the  smallest  number  that  will  exactly  contain 
16,  24,  and  30? 

30.  How  long  must  a  box  be  that  no  room  may  be  lost  in 
packing  in  it  books  6  inches,  8  inches,  or  12  inches  long? 

31.  A  lady  desires  to  purchase  a  piece  of  cloth  that  can 
.be  cut  without  waste,  into  parts  4,  5,  or  6  yards  long.     How 

many  yards  must  the  piece  contain? 

32.  I  have  a  certain  number  of  pennies  which  I  can  ar- 
range in  either  4,  6,  8,  10,  or  12  equal  piles.  What  num- 
ber of  i">ennio6  have  I,  if  it  is  tho  least  number  tliat  admits 
of  such  arrangement? 

33.  How  many  bushels  will  the  smallest  bin  contain  that 
can  be  emptied  by  taking  out  oitlicr  7  l)iish('ls,  10  bushels,  or 
30  bushels  at  a  time  ? 

34.  Four  agents  start  from  2sew  iurk  at  the  same  time. 
The  first  makes  his  trip  in  8  weeks,  the  second  in  9  weeks, 
the  third  in  15  weeks,  and  the  fourth  in  20  weeks.  How 
many  weeks  will  pass  by  before  they  will  again  start  out  from 
New  York  together? 


92  PIIOPEIITIJ        i)i      NUMBERS. 

35.  Three  men  walk  around  a  circular  inland,  the  circum- 
ference of  which  is  360  miles.  A  \  alk-  1 "»  miles  a  day,  B 
18  miles  a  day,  and  C  24  miles  a  day.  it"  they  start  together 
and  walk  in  the  same  direction,  how  manv  days  will  elapse 
iKjfore  they  will  be  together  again  ? 

36.  Divide  5  X  15  X  80  X  56  X  81  by  lu : ;  o  A  16  X  78. 

37.  If  a  man  buys  a  lot  whose  sides  measure  respectively 
48  feet,  60  feet,  96  feet  and  108  feet,  what  will  be  the  length 
of  the  longest  IxMirds  which  he  can  use  to  fence  all  the  sides 
without  cTtttiug? 

3>  he  greatest  common  divisor  of  1744,  9564  and 

8524. 

39.  What  is  the  smallest  number  which  can  be  divided  by 
250,  350,  and  525  resijectively ,  and  leave  a  remainder  of  25  ? 

40.  What  is  the  greatest  common  divisor  of  ^"■'"  ^"'12, 
and  8640? 

41.  A  stock  buyer  wishes  to  invest  the  same  amount  of 
money  in  sheep  at  %Z  each,  hogs  at  $14  each,  and  cows  at  821 
each,  as  he  does  in  beef  cattle  at  $48  each.  What  is  the 
smallest  {Mssiblc  amount  which  he  can  invest  in  each  ? 

42.  Jones  Brothers  &  Co.,  of  Cincinnati,  O.,  received  an 
order  for  a  numlKjr  of  Lyman*s  Historical  Charts.  It  was 
found  that  if  the  charts  were  j>acked  in  boxes  containing 
cither  24,  28,  32,  or  36  charts  each,  there  was  a  remainder 
of  9  each  time,  but  if  packed  in  boxes  containing  25  each, 
there  was  no  remainder.     How  many  charts  were  onlered? 

43.  A  dealer  in  real  estate  purchased  3  lots  of  land  whose 
width  on  the  street  were  res|)ectively  152  hkIs,  288  rods,  and 
184  rods.  What  is  the  width  of  the  largest  lots  of  equal  size 
which  can  l)e  formed  from  them  ? 

44.  Divide3x5x20Xl0x3xl3  by  26x9x3x4. 

45.  Find  the  greatest  common  divisor  of  2219,  4501,  and 
5964. 

46.  Divide 5x8x3x7X 28 X99 by  11x4x7x5x4. 


onelmir 

.. 

onc-third       1            % 

H 

ODe>rourth 

h 

M 

'4 

one- 

nrth 

% 

Vs 

'A 

« 

What  are  three  of  the  parts 


140.  1.  When  a  line  is 
divided  into  two  equal  parts, 
what  is  each  j)art  called? 

2.  When  a  line  is  divided 
into  iJiree  equal  parts,  what 
is  each  part  called?  What 
are  two  of  the  parts  called? 

3.  When  a  line  is  divided 
into  four  equals  parts,  what 
is  each  part  called?     What 
are  two  of  the   parts  called? 
called? 

4.  WliLn  aii\  lliiug  is  divided  into  five  equal  parts, 
what  is  each  part  called?  What  are  three  parts  called? 
What  are  four  parts  called? 

5.  When  things  are  divided  into  6,  7,  8,  9,  10,  15  equal 
parts  respectively,  what  is  each  of  the  parts  called  ?  What 
are  four  of  them  called  ? 

6.  How  many  halves  are  there  in  any  thing?  How  many 
thirds?     Fourths?     Fifths?    Sixths?    Tenths? 

7.  If  10  marbles  are  separated  into  5  equal  groups,  what 
part  of  the  marbles  will  l)e  in  each  group  ? 

8.  How  many  are  one-fifth  of  10?  Two-fifths?  Three- 
fifths? 

9.  How  many  are  one-sixth  of  12?    Two-sixths? 

(83) 


94  COMMON    FKACTION8. 


DEFINITIONS. 

141.  A  Fraction  is  one  or  more  of  the  equal  parts 
of  a  unit. 

142.  The  Unit  of  a  Fraction  is  the  unit  which  is 
divided  into  equal  parts. 

A  fraction  in  which  the  unit  has  been  divided  into  any  number  of 
equal  partH  is  called  a  Common  Vraction, 

A  fraction  in  which  the  unit  has  been  divided  into  fejiiLt,  hun- 
dndlhMf  tkoummdUUf  etc.,  is  called  a  Decimal  Frtiction. 

143.  A  Fractional  Unit  is  one  of  the  equal  parts  into 
which  a  unit  b  divided. 

144.  Since  a  fraction  is  one  or  more  of  the  equal  parts  of 
any  thing,  to  express  a  fraction  two  numbers  are  necessary, 
one  to  express  the  number  of  equal  parts  into  which  the  unit 
has  been  divided,  the  other  to  express  how  many  make  the 
fraction.  These  numbers  are  written  one  above  the  other 
with  a  horizontal  line  between  them. 

145.  The  Dcnoniinator  is  the  number  which  shows 
into  how  many  equal  parts  the  unit  is  divided. 

It  is  written  below  the  line. 

Thus,  in  the  fraction  ^ ,  7  is  the  denominator.  It  shows  that  the 
unit  of  the  fraction  has  been  divided  into  7  equal  parts. 

146.  The  Numerator  is  the  number  which  shows  how 
many  fractional  units  form  the  fraction. 

It  is  written  above  the  line. 

Thus,  in  the  fraction  7,  5  Ls  the  numerator  and  shows  how  many 
fractional  units  form  the  fraction. 

147.  The  numerator  and  denominator  are  called  the 
Terms  of  a  Fraction. 


148.  Fnu'liuiial  units  are  named  from  llie  miinber  of  puru 
into  which  the  unit  is  divided.  Thus,  ^  is  read  one-sixth^; 
■f ,  one-sevetUh. 

Fractions  are  read  by  naming  the  number  and  kind  of  frac- 
tional units.  Thus,  J  is  read  five-sixths ;  ^^,  five  twenty-firsts ; 
U,  thirteen  thirty-fifths. 


U9. 

Read  the  following: 

A 

H     ^     if 

if 

« 

w 

m 

^    ^    ^ 

T«*T 

106600 

ASWW 

m 

Aia.         469         8g§ 
TTT        TsT        Wt 

ttM 

imi 

fWWr 

Express  by  figures: 

1.  Three   elevenths.     Five   thirteenths.     Eiglit    twenty- 
firsts. 

2.  Forty-eight  fiftieths.     Twenty-seven  eighty-fifths. 

3.  Sixty  forty-eighths.     Fifty-seven  ninety-ninths. 

4.  Forty-two  eighty-sevenths.     Thirty-nine  ninety-thirds. 

5.  Seventy-four  one-hundredths.      Ninety-seven  one-hun- 
dred-fifths. 

6.  Fifty-two  seventy-eighths.     Tliirty-six  eighty-fourths. 

7.  Two  hundred  three-hundred-ninetieths. 

8.  Seven  hundred  seventy-one  eight-hundred-sixtieths. 

9.  Two  hundred  forty-nine  three-hundredths. 

10.  Five  hundred  sixty-six  seven-hundred-fiftieths. 

11.  One  hundred  eleven  two-hundredths. 

12.  Four  thousand  six  hundred  thirty  five-thousandths. 

Fractions  are  classified  with   reference   to  the   relation  of 
numenitor  and  denominator  tluis: 

150.  A  Proper  Fraction  is  one  in  which  the  numer- 
ator is  less  than  the  denominator. 

Thus,  J,  §,  ||l,  etc.,  are  pro|H'r  fractions. 

The  value  of  a  proiier  fraction  18  therefore  less  than  1. 


96 


CO>fM()N'    FRACTIONS. 


1.>1.  An  liiipi'opvi'  J:yactlon  is  a  fraction  in  wliicli 
tlie  numerator  equals  or  exceeds  the  denominator. 

Thus,  },  I,  \Z,  are  improper  fractions. 

The  value  of  an  improper  fraction  is  therefore  1  or  more  than  1. 

152.  A  Mixed  yintiher  is  a  nu>'»l»"r  "xpressed  by  an 
integer  and  a  fraction. 

Thus,  2 1,  5},  are  mixed  numbers. 

Mixed  numbers  are  read  by  naming  the  fraction  after  the 
whole  number.     Thus,  2J  is  read  two  and  tJiree-fourUis. 

Fractions  may  be  regarded  as  expressing  unexecuted  ^//ivV 
ion.    Thus,  ^  is  equal  to  16-4-8  ;  Y  '^  ^^^^  15-5-'). 

153.  1.  Interpret  the  expression  -f. 

Analysis. — ^  represents  5  of  7  equal  part.*?  into  which  any  thing  is 
divided.  It  also  represents  one-eeviuth  of  flvf,  miuI  o  dividid  by  7. 
It  is  read  jite-aaxniha. 

In  like  manner  interpn  i :     • 


2.  f 

3.  T«r- 

4.  A- 


6.  «. 

6.  a- 


8-  m- 

9-  AV 

10.  W- 


11.  SJ. 

12.  U- 

13.  ^. 


REDUCTION 


CASE    I. 
154.  To  rcdare  fVnrllons  lo  larger,  or  higher  terms. 

1.  In  -i-  of  an  apple  how  many  fourths  are  there?  How 
many  eighths  ? 

2.  How  many  sixths  are  there  in  ^?  How  many  ninths? 
How  are  the  terms  of  the  fraction  f  obtained  from  those  of 
Jf     I  from  i? 

3.  How  many  eighths  are  there  in  J?    How  many  twelfths? 


UKDlKmON.  !)< 

4.  llv>\v  <lo  liic'  Uriii-  nf  ilir  iVariidii   ';    cMnijjan'  \\\[\i  the 
terms  of  the  fraction  \ '! 

5.  In  what  equivalent  fraction  can  \  be  expressed  ? 

6.  How  do  the  terms  of  tlic    fraction    |    com))are   with 
those  of  t\? 

7.  How  are  the  terms  oi    liu-  iiaction  -j-^^  ol)taincd  I'rom 
those  of  \  ? 

8.  How  are  the  terms  of  the  fraction  |-  obtained  from  J? 

9.  How  are  the  terms  of  the  fraction  f  obtained  from  ^? 
10.  What  then  may  be  done   to  the  terms  of  a  fraction 

without  changing  the  value  of  the  fraction  ? 


11.  Change  \  to  24ths. 

12.  Change  f  to  16ths. 

13.  Change  f  to  24ths. 


15.  Change  -f*^  to  36ths. 

16.  Change  f  to  20ths. 

17.  Change  ^  to  14ths. 


14.  Change  J  to  12ths.  1^.  Change  |  to  18ths. 

155.  llefluctlon  of  I'i'actions   is   the   process  of 
changing  their  form  without  changing  their  value. 

156.  A  fraction  is  expressed  in  Larger  or   H  iff  her 

Terms  when  its  numerator  and  denominator  are  expressed 
by  larger  numbers, 

157.  Principle. — MuUipbjing  both  terms  of  a  fraction  by  the 
name  number y  does  not  change  tJie  value  of  tlie  fraction. 


WRITTEN     1    V  /    i:  <   I  s  /    s 


1.  Cliange  ^  to  forty-fifths. 


Analysis. — Since  there  are  4  j  forty-liithrt  in 

45-4-15  =  3         It  in  T  5  there  are  3  forty -fifths;  and  in  /^  there 

are  7  times  ^V,  or  J]^;  or, 

-  '  Since  the  denominator  of  the  required  frac- 

1  >        '■',        1")  tion  is  3  times  that  of  the  given  fraction,  we 

must  multiply  the  terms  of  the  fraction  by  3. 


98 


COMMON    FRACTIONS. 


Rule. — Multiply  Vie  terms  of  Hie  fraction  by  such  a  number  as 
will  diange  the  given  denominator  to  tJie  required  denominator. 


Reduce: 


2.  ^  to  50th8. 

3.  ^  to  eoths. 

4.  Il  to  TOths. 

5.  }|  to  84ths. 

6.  If  to  40ths. 

7.  1^  to  54ths. 

8.  II  to  66ths. 

9.  If  to  54ths. 
10.  II  to  84tb8. 


Reduce: 

11.  J4  to  120ths. 

12.  II  to  64th8. 

13.  1^  to  74th'8. 

14.  ^  to  210th8. 

15.  II  to  240th8. 

16.  fl  to  22511.8. 

17.  fj  to  348th8. 

18.  II  to  558th8. 

19.  1}  to  235tha. 


CASE    II. 
158.  To  reduce  frartioufi  (o  smaller,  or  lower  ferms. 

1.  How  many  fourths  are  there  in  J?     How  many  in  -j^? 

2.  How  many  ihirdi^  are  there  in  J?    How  many  in  ^? 

3.  How  does  the  nunilxjr  of  eighths  of  any  thing  compare 
with  the  fourths?    Thirds  with  sixtlis?    Halves  with  eights? 

4.  How  do  the  terms  of  the  fraction  |  comj^are  with  tlio<f« 
of  I  ?     How  with  those  of  -,%? 

5.  How  do  the  terms  of  the  fnutiiiii  \  ,.,,  ;,,.   . 
of  I  ?     How  with  those  of  -f^l 

6.  How  are  the  terms  of  the  fraction  \  uhtiiiueil  from  those 
of  the  fraction  J?    How  from  those  of  yV? 

7.  How  are  the  terms  of  the  faction  |  obtained  from  |? 

8.  What  then  may  be  done  to  the  terms  of  a  fraction 
without  changing  the  value  of  the  fraction? 

9.  Express  y\,   -j^,    -fir,  in  smaller  or  lower  terms. 

10.  Express  -J-J,   If,    |f ,   in  smaller  or  lower  terms. 

11.  Reduce  -^,   IJ,    J^,   to  smaller  or  lower  terms. 

12.  Reduce  |^,  -j^^,  ^^,  to  smaller  or  lower  terms. 


RKDUCTION.  ^»^» 

159.  A  fnictioM  is  expressed  in  Sniff ffer^  or  Liner r 
Terms  wlien  its  iiumcnitor  and  denominator  are  expressed 
in  siiicUler  numbers. 

KJO.  A  fraction  is  expressed  in  tlie  Stnaffesf^  or 
LiOWest  Terms  when  its  numerator  and  denominator 
have  no  common  divisor. 

161.  Principle. — Dividing  hoUi  terms  of  a  fraction  by  the 
same  mimber  does  not  change  Vie  value  of  Uie  fraction. 


WniTTBN    EXEJICISES. 

16'2.  1.  Change  }f  to  an  equivalent  fraction  expressed  in 
its  smallest,  or  lowest  terms. 

.  Analysis. — Since  tlie  denominator  of 
the  required  fraction  is  to  be  smaller  than 
that  of  the  given  fraction,  we  may  obtain 
an  equivalent  fraction  having  smaller 
terms,  by  dividing  the  terms  of  the  given 
fraction  by  any  exact  divisor,  aa  4  (Prin.), 
and  the  terms  of  the  resulting  fraction  by  4. 
We  thus  obtain  the  fraction  |,  whose  terms 
have  no  common  divisor.    The  fraction  is 

therefore  in  its  smallest  terms.     Or, 

Since  fractions  are  in  their  Rmalle8t  terms  when  their  numerator 

and  <U'nominator  have  no  common  divisor,  to   reduce  them  to  their 

smallest  terujs  we  may  divide  both  terms  by  their  greatest  common 

<li  visor. 

Rule. — Divide  the  numerator  and  denominator  by  any  common 
divisor^  and  continue  to  divide  fhiofvvtif  the  frrms  have  no  common 
divisoTy  Or, 

Divide  botJi  terms  by  tic n  yuaif.-i  luntmun  liii'isor. 

2.  Reduce  -J^,  ^J,  ^|},  |^,  to  their  smallest  terms. 
X  Re<luce  }j,.  -ftj^  ih  +H»  *"  their  smallest  terms. 


PROCESS. 

4  32  _4 
4  48     4 

8  _2 
12  —  3 

Or, 

32      32  -- 16      2 

'48      48- 

-16      3 

100  COMMON    FRACTIONS. 

Reduce  to  their  smallest,  or  lowest  terms: 


4.  H. 

5.  H. 


9.  m- 

10.  Hi- 

12.  m- 

13.  tSSV. 


14.  m- 

15.  tW. 

16. +m. 
17.  im. 

18.  HH. 


19. 
20. 
21. 


M4i 

8808' 
694»' 

JOJUL 


22.  tViV,. 

23.  tH*^. 


CASE  IIL 

!(»:>.    Eo    r<^nre    iiitt^Ker^i    or    mixed     munbeni    to 
improper  flVartionit. 

1.  How  many  halves  are  there  in  1  apple  ipples? 
In  6  apples? 

2.  How  many  thirds  are  there  in  1  orange?     In  :)  <.raii::«  -7 
In  5  oranges? 

3.  How  many  lourihs  are  there  in  2?     In  3?     In  4? 

4.  How  many  fifths  are  there  in  3?    In  4?    In  6? 

5.  How  many  fourths  are  there  in  IJ?    In  2|? 

6.  How  many  thirds  are  there  in  2}?    In  M  ?    In  6J? 


WKITTXN    BXKRCrsES. 

104.  1.  l\educe  8f  to  sevenths. 

PROCESS.  Analysis. — Since   in    1    there     are  7 

8  =  4A  8eventh.s,    in    8    there     are     8     times    7 

"  sevenths,  or  ^;  and    in  ^      ^    tb'-*"   niv 

8f  =  ¥  +  f  =  ¥        ^+hor>^. 

Rule.— 3/»/<i;)/y  iJie  integers  by  the  given  dejiominator,   to 
this  product  add  tJie  numerator  of  the  fractional  party  if  there  be 

anv,  and  wriff'  the  r''>'"^'  ' -  '^'"  '••■•''»  denominator. 


2.  Change  b\  to  loiirni: 

3.  Change  15  to  fifths! 


4.  Reduce  13J  to  sixths. 

5.  Reduce  18^\  to  elevenths. 


RED  U  en  O.N. 


101 


w.  *.  iiiuige  5 J  to  ninths.     To  eightecntiis. 

7.  Chanji^e  (iiV  to  twelftjis.     To  twenty -fourths. 

s.  Change  8y\  to  fourteenths.     To  forty-seconds. 

[).  lieduee  9./(^  to  twentieths.     To  sixtieths. 

Reduce  to  improper  fractious: 


10.  13f. 

11.  12f 

12.  18^. 

13.  23f. 


14.  25f 

15.  293^. 

16.  37|}. 

17.  56}f. 


18.  421^. 

19.  540}-|. 

20.  763^. 

21.  419yV 


22.  867^. 

23.  904Jf. 

24.  Sl'^. 

25.  721-j>2^. 


( Asi:  IV. 

105.  To   rodncc  improper   IVactions   to   inte[a;crs  or 

lIliXCMl   llUllllM*rM. 

1.  How  many  days  are  there  in  6  half-days?     In  8  half- 
days?     In  14  luilf-<lays? 

2.  How  many  yards  are  there  in  9  thirds  of  a  yard?    In 
15  thirds?     In  18  thirds? 

3.  If  a  boy  pick  ^-  bushel  of  peaches  per  hour,  how  many 
bushels  can  he  pick  in  10  hours?   How  many  are  10  lialves  ? 

^.  If  a  man  can  earn  \  of  a  dollar  per  hour,  how  much 
can  he  earn  in  12  hours?    How  many  are  12  fourths? 

5.  How  many  units  are  there  in  J/?    ^?    ^?    J^-?    ^? 
().  How  many  dollars  are  there  in  §3^?   $^?   $J^?   $^^? 


WJtJTTEN    EXERCISES, 

166.    1.  Reduce  ^^  to  a  mixed  number. 

PROCESS.  Analysis. — Since  7  sevenths  equal 

LIZ  —  123  -^  7  =  174.        1  unit,  123  sevenths  are  equal  to  as 
many  units  as  7  sevenths  arc  contained 
n  r  17^  times.     Therefore  if  a  =  17^. 

Kn.i;.  —  I>imde  the  numerator  by  tiie  denominator. 


102 


COMMON   FRACTIONS. 


2.  Lhan-e  8^  to  dollars.  • 

3.  Change  -^j^  pounds  to  pounds. 

4.  Clmngc  ^jV  ounces  to  ounces. 

5.  Change  ^|^  to  a  mixwl  number. 

6.  Change  ^  to  a  mixed  number. 

7.  Change  ^  to  a  mixed  number. 

lieduce  to  integers  or  mixed  numbers: 


8.   W- 

12.  i^. 

16.    ^Vtf. 

20.  HHJ. 

9-   W- 

13.  ^iW- 

17.  Htt*. 

21.  W?f 

10.  w- 

14.  W/ 

18.  Hlt^. 

22.  WiV 

11.  VA*- 

15.  W/. 

19.  HH*. 

23.  WiV. 

cam:  v. 

167.  To  rodiieo  diMHliuilar  fraotioiiN  to  Niniilar  frac* 

tiOIIM. 

1.  How  many  fourths  are  there  in  ^  of  an  orange? 

2.  How  many  »ij^/w  of  a  field  are  there  in  ^  of  a  field? 

3.  How  many  eighth  in  J?     How  many  niMs  in  J? 

4.  Express  each  of  the  fractions  J,  },  and  |  as  twelfths. 

5.  Expres.s  each  of  the  fractions  ^  and  f  as  twentieths. 

6.  If  \  is  divided  into  3  equal  parts,  how  large  is  each  part? 

7.  If  J  is  divided  into  2  equal  parts,  how  large  is  each  pa^rt? 

8.  When  J  and  ^  are  divided  into  equal  |)arts,  what  parts 
are  common  to  both  ? 

9.  When  J  and  ^  are  divided  into  .  lual  part?,  what  jarts 
are  common  to  both  ? 

10.  What  equal  parts  are  common  to  both  \  and  ^? 

11.  AVhen  \,  \  and  |  are  divided  into  etjual  parts,  ^m.ai 
parts  are  common  to  all? 

12.  CJiange  ^  ,  J,  i,  to  equivalent  fractions  having  the  same 
fractional  unit.  Express  the  resulting  fiiictions  in  equivalont 
fractions  having  their  least  common  denominator. 


T?  EDUCTION.  lUo 

Reduce  to  fractions  liuving  the  same  fractional  unit: 


11.  Jand  f 

12.  I  and  -ft. 

13.  i  and  f 


14.  f  and  ■^. 

15.  f  and  t^. 

16.  I  and  -j^. 


17.  A  and  -ft. 

18.  tV  and  ■^. 

19.  ^  and  ^. 


168.  Similar  Fractions  are  those  that  have  the  same 
fmetional  unit. 

169.  T>issiinilar  Fractions  are  those  that  have  not 
the  siime  fractional  unit. 

170.  Similar  fractions  have  a  Common  Denomi- 
nator, 

171.  When  similar  fractions  are  expressed  in  their  small- 
ed  terms  they  have  their  Least  Common  Uenomir- 
nator, 

172.  PiaNciPLES. — 1.  A  common  denomhmtor  of  two  or  more 
fractions  is  a  common  mtdtiple  of  tJieir  denominators. 

2.  The  least  common  denominator  of  two  or  more  fractions  is 
tlie  least  common  mxdtiph  of  their  denominators. 


wniTTBN  jf:xeiicises. 

ViW,    1.   luthice  {  and  f  to  similar  fractions. 

PROCESS.  Analysis. — Since  similar  fractions  have  a 

^—-^y.S—-2A         common  denominator,  to  make  these  fractions 
pimilar  wc  must  change*"  them   to  equivalent 
T ^^ ¥^  4  ^^83         fractions  liaving  a  common  denominator. 

Since  a  common  denominator  of  two  or 
more  fraction;*  \a  a  eonimon  multiple  of  their  denominators  (Prin.), 
wu  find  a  commcm  niiiltiplt-  of  tlio  denominators  1  and  S,  wliitli 
is  32. 

We  then  multiply  the  terms  of  each  fraction  hy  such  a  number  as 
will  change  the  fraction  to  thirty-seconds. 


101 


COMMON    I  l;  \(    1 


li.     ilLd.,,^ 


..i 


iiasiiii:    lluir 


least  common  denominator. 


rnociss. 


tion  by  611  !iiilx?r 

vrhosc  denominator  l»  I'J 

Since  1  ib  equal  to  |n, 

to  2  timefl  ^j,  or  ^^,  etc. 


Analysis. — ^The  least  common  denomina- 
tor of  several  fractions  is  the  least  common 
multiple  of  their  denominators  (Prin.  2); 
therefore  we  find  the  least  common  nailtiplc 
of  3,  4,  and  G,  which  is  12.  • 

We  then  multijtly  the  terms  of  each  frac- 
as   will   fhllllLT    it    t«»   hiih'lliA     nr     Im'  :i    f  r:ic(  i.  in 


•  iual 


Rule. — Find  Hie  common^  or  I^aM  common  multiple  of  (he 
deiiominaionfor  a  common,  or  !  non  denominator. 

Divide  this  denominator  by  tJu  >•> mnninator  of  each  frad'wn 
ami  multii)ly  both  terms  of  tlie  fraction  by  the  quotient. 

Reduce  all  mixed  numbers  to  improper  fractions  and  all  fractions 
to  their  smallest  terms. 

Change  tlie  following  to  simibr  fractions  having  their  least 
common  denominator : 


3.  h  i.  A- 

7.  f  4.  A- 

11.  3?.  If  if. 

4.  i.  ii.  «. 

8.  *.  ^.  A- 

12.  a,  n,  u- 

5.  I.A.H- 

9.   1.  «.  «• 

13.  H. «.  if 

6.  f  A.  «• 

10.  A,  \i, «. 

14.  H,  A.  H- 

ADDITION. 


17-4.  1.  James  has  2  fifths  of  a  dollar,  and  his  brother 
has  4  fifths  of  a  dollar.     How  many  fifths  have  both? 

2.  George  sjient  S^  on  Monday,  and  $1  on  Tuesday.  How 
much  did  he  spend  in  both  days?  How  many  sevenths  are  ^ 
and  i? 


3.  Mr.  A.  .sold  I  of  his  f'tirm  at  one  time,  and  J  at  another 
time.  What  jmrt  of  it  did  he  sell  ?  What  is  the  sum  of  J- 
and  J? 

4.  James  caughl  «  li.sh  in  the  niorniug  that  weighed  J  of  a 
jwund,  and  another  in  the  afternoon  that  weighed  ■}  of  a  i)ound. 
What  did  hoth  weigh?     What  is  the  sum  of  |  and  J? 

5.  Marian  gave  $J-  for  a  book  and  $^  for  some  writing 
l)ai)er.  How  much  did  she  pay  for  both?  What  is  the  sum 
of  >  and  } ? 

G.  Ella  gave  }  of  her  apple  to  a  poor  beggar  and  Julia 
gave  him  J  of  hers.  How  many  fourths  did  he  receive? 
What  is  the  sum  of  J  and  -J? 

7.  I  bought  }  of  an  acre  of  ground  for  a  site  for  a  house, 
and  li  of  an  acre  for  a  site  for  a  barn.  How  much  land  did  I 
buy?  What  is  the  sum  of  ^  and  J^  ?  Of  -^  and  J?  Of  | 
and^^^? 

8.  Mr.  A.  gave  8fV  to  one  man  and  $J  to  another.  How 
much  did  he  give  to  both? 

9.  A  merchant  sold  J  of  a  bushel  of  clover  seed  to  one 
farmer,  and  |  of  a  bushel  to  another.  How  much  did  he 
scU  to  Iwth  ? 

10.  Sarah  paid  8|  for  eggs  and  $J-  for  butter.     How  much 
did  Ix^tli  cost  her? 

11.  1  i)aid  $1  for  turnips  and  $|  for  squashes.    How  much 
ditl  1  i)ay  for  both  ? 

12.  A  merchant  sold  J  of  a  yard  of  silk  to  one  lady  and  J 
of  a  yard  to  another.     How  much  did  he  sell  to  both? 

13.  A  boy  earned  S^  in  the  forenoon  and  3J  in  the  after- 
noon.    How  much  did  he  earn  that  day? 

14.  What  is  the  sum  of -J  and  -jV?    f  and  /j?    yV  «"^^  ^u'- 
Id.   What  kind  of  fractions  c^m  ho  nddcd  wiflionf  <'hnn«nnjr 

their  form? 

16.  What  must  bo  done  to  dis-imiJur  Inictious  hclure  they 
am  be  added?     How  are  dissimilar  fmctions  made  similar? 


IOC  CX>MMON    FU ACTIONS. 

175.  ri:i.NciiLE8. — 1.   Only  similar  fractions  am  be  added. 
2.  Dimmilar  fractions  mud  be  reduced  to  similar  fraetiom 

before  adding. 

WR I  i  I  I    \      I    \   I    !:  '    1  \  IS, 

176.  1.  What  is  tho  sum  of  #,  f  and  |? 

PROCESS.  Ana  lysis.— Since  the  frac- 

A-}.  &-[.l=:AQ.-|.Jl.-|.JL=:^    Uons  arc  not  Bimilar,  before 

adding  we  must  change  them 
to  giniilor  fractions,  or  equivaleiit  fractrons  having  a  common  denom- 
inator. 

The  least  common  denominator  of  the  given  fractions  is  3fi;  and 
|  =  {g,  f  =  li»  anJ  w^s's*  Hence  the  sum  of  the  given  frnctions 
must  bo  equal  to  the  sum  of  |g,  ||,  and  ^^  which  ia  fl,  or  Ifg. 

2.  W.hat  is  the  sum  of  5},  6}  and  2f  ? 

TROCEsa.  Analysis. — Since  the  numbers  are  composed  of 

51  —  5JL.  both  intf^rs  and  fraction)*,  we  may  add  each  Kei>- 

^, Aj,  arately  and  unite  the  sums.    Thus,  the  sum  of  tlic 

«       ^^  fractional  jiarts  is  \\^  or  \\\\  the  sum  of  (In  inu- 

^}  =  ^lV  g^i^  i^  13;  »°^  *i*<^  ^""*  ^^  ^'^'i>  I'^H- 

KuLE. — Reduce  the  given  fraction  unlar  fractions^  add 

their  numerators  and  write  Hie  suni  ovei'  the  common  denomi- 
nator. 

When  tliere  are  mixed  numbers,  or  integers,  add  Hie  fractions 
and  integers  sqtarately  and  Hien  add  tJte  results. 

If  tlie  sum  be  an  improper  fraction,  reduce  to  an  integral  or  mixed 
number. 

Find  the  sum 

3.  Of  I  ^,  J,  5  and  H- 

4.  Of  i,  i,  -\,  i  and  f 

5.  Of  I  i,  I  i  and  ^. 
C.  Of  {,  },  ^,  i  and  A. 


7. 

Of  §,  H,  H  and  U. 

8. 

Of  2Jlr,  4,  3 J  and  5|. 

9. 

Of  27^,  8|  and  40|. 

10.  Of  13f ,  15|  and  20J|. 


■I  i;l  KACTIOX.  107 


Add  the  following: 

11.  h  M.  if.  /..  A- 

12.  4f,  5f,  8i,  2f,  7i,  4f. 

13.  9t,  7J,  8J,  7A.  8^. 


14.  7tV,  8,  6^,  3^,  5^. 

15.  fi,  M,  2A.  3A,  2i|. 

16.  3f,  4A,  6.  9A,  A- 


17.  A  farmer  received  $18|  for  hay,  665f  for  a  cow,  and 
SI 61}  for  a  horse.     How  much  did  he  receive  for  all? 

18.  A  man  earns  ?67f  per  mouth,  and  each  of  his  two 
sons  $23}  per  month.     How  much  do  all  earn  per  month? 

19.  A  pedestrian  walked  45J  miles  on  Monday,  47f  on 
Tuesday,  50^  on  Wednesday.     How  far  did  he  walk? 

20.  A  has  5\  acres  of  land,  B  has  lOf  acres  more  than  A, 
C  has  as  much  as  both  A  and  B.  How  many  acres  have  B 
and  C  together? 

SUBTRACTION. 

177.  1.  ISfary  earned  5  ninths  of  a  dollar  and  spent  2 
ninths.     How  many  ninths  of  a  dollar  had  she  left? 

.2.  Mr.  A.  owning  ^  of  a  flouring  mill,  sold  ^  of  it.  How 
many  sevenths  did  he  then  own? 

3.  From  |  subtract  f.  From  }  subtract  |.  From  -jfij- 
subtract  -^. 

4.  From  |J  subtract  ^^.     From  -^  subtract  -j^. 

5.  Mr.  B.  owned  a  lot  containing  J  of  an  acre.  How  much 
had  he  left  after  selling  ^of  an  acre? 

6.  A  boy  paid  SJ  for  a  whip,  but  sold  it  after  a  time  for 
$J.     How  much  did  he  lose? 

7.  Find  the  difference  l)ctween  ^  and  J.     J  and  J. 

8.  What  kind  of  fractions  can  be  subtracted  without 
changing  their  form? 

9.  What  must  be  done  to  di.s><imilar  fractions  bcfoiv  tiiey 
can  be  subtra'cted ?    How  are  dissimilar  fractions  nuide  similar? 


108  COMMON    FRACTIONS. 

178,  Principles. — 1.  (My  similar  fradunu  can  he  mhtraded, 
2.  IHmmilar  fradions  mud  be  reduced  to  similar  fracU&ns 

before  subtractXtig, 

WBITTBN    JBXBBCI8B8, 

179.  1.  Wliat  is  the  difference  bet  wren  |4  and  |? 

PROCESS.  Analy8I>  lic  fractioiw  nro  not 

1± 1=^-|1 A-        Mmilar,  bcforv  subtracting  we  must  change 

them  to  Hiniilar  fractions. 
Ir^^^^^Tf  '       'onimon  denominator  of  the  given 

IIS  w  [i6;  ami  1*  =  ^!  tt"^  l  =  A' 
Henoe  the  diflerenoe  between  the  given  fractions  is  equal  to  the  dif- 
ference between  |}  and  ^,  which  is  ]}. 

2.  What  is  the  diffomicc  between  23J  and  4#. 

PROCESS.  Analysis. — Since  the  numbers  are 

231.  =  23-A-  =  22  '  '<>nipo«cd  of  both  integers  and  frac- 

41=    4n  =  JM       S.."  ""  ^"■"  ''^"  """" 

^^tV  ^^^^  ^•"'^  reduce  tlic  given  fractions 

to  similar  fractions.  Since  we  can 
not  take  {}  from  -ff^  we  unite  with  the  i'^  1,  or  j^,  taken  from  23, 
making  \l.    Then  22J| ~  4^  =  18/^,  the  remainder. 

Rule. — Reduce  the  fractums  to  similar  fractions. 

Find  Uie  difference  cfihe  numerators  and  write  it  over  the  com- 
man  denominator. 

Wlien  tliere  are  mixed  numhers  or  integers^  subtract  tJie  frac- 
tions and  integers  separatdy. 

Mixed  nuniliers  may  be  reduced  to  improper  fractions  and  sub- 
tracted according  to  the  first  part  of  the  rule. 


(3.) 

(4.) 

(5.) 

(6.) 

(7.) 

(8.) 

From    i 

* 

A 

« 

H 

^ 

Take     f 

A 

A 

^ 

*i  * 

A 

MI  I  rii'i 

u 

9. 

1        tWM.                -                 l.t...              Jy. 

10. 

From    ^   take  |. 

11. 

From  ^%  take  -j^j. 

12. 

From  {'^  tiike  -^g. 

13. 

From  iJ  take  tjV 

14. 

From  IJ  take  f  j. 

•ATION.  100 

10.  iVom  lOy  take  ,;,.j. 

16.  From    C6J  take  33». 

17.  From  210^  take  10!)^. 

18.  From  112    take  75. V. 

19.  From  606|  take  70i. 

20.  From  589 j  take  67]. 

21.  If  from  a  bin  containing  506j  tons  of  coal,  418J  tons 
are  taken,  how  many  tons  still  remain? 

22.  A  lady  having  $25,  paid  §2  J  for  a  pair  of  gloves,  $15| 
for  a  Iwnnet,  and  $3J  for  some  lace.  How  much  money  had 
she  left? 

23.  A  man  owned  a  farm  of  412  acres.  He  sold  three 
parcels  of  land  from  it,  the  first  contiiining  60|  acres,  the 
second  45 .J^  acres,  and  the  third  116  J  acres.  How  many  acres 
did  he  sell,  and  how  many  had  he  remaining? 

24.  A  clerk  earned  §50^  i>er  month.  He  paid  S20J  for 
board,  S5J  for  washing,  and  $4^  for  other  exi)enses.  How 
much  did  he  save  per  month? 


MULTIPLICATION. 

CASE  I. 
180.  To  iiiiilliplj  a  fVaciioii  hy  an  integer. 

1.  At  $\  a  yard  wliat  will  3  yards  of  cambric  cost? 

2.  If  a  man  can  earn  $^  per  hour,  how  much  can  he  earn 
in  5  hours?    How  much  can  he  earn  in  8  hours? 

3.  James  gave  }  of  an  apple  to  each  of  5  cluldren.     How 
many  apples  did  he  give  to  all?     How  much  is  5  times  J? 

4.  How  many  fifths  are  there  in  6  times  J?     In  7  times  f  ? 

5.  If  Mr.  A.  spends  $2^  per  day,  how  much  will  he  spend 
in  5  days?     How  much  in  10  days? 


110 


COMMON    FRACTIOXS. 


G.  How  much  is  2   times  f  ?     How  (ioe<  ihc  ivsuk  c(jni- 
pare  with  |?    How  is  it  obtained  from  J? 

7.  In  multiplying  a  fraction  what  part  of  the  fraction  do 
we  multiply? 

8.  Multiply  l  l.y  '2.     i  by  3.     ^«r  by  7. 

9.  Express  2  times    J  in  smallest  terms.     How    is   this 
result  obtained  from  the  fraction  f  ? 

10.  In  what  other  way  then  may  we  multiply  a  fraction? 

11.  How  much  is  3  times  |?     4  times   J  ?     6  times  -j%? 

12.  How  much  is  5  times  ^?     6  times  |  ?     9  times  J  ? 

13.  How  much  is  4  times  -f?    3  times  ^2    5  times  |-  ? 

14.  How  much  is  4  times  4?     7  times  ^?    9  times  ^? 

181.  Principle. — Mukiphjing  (he  numerator  or  dividing  the 
denominator  of  a  fraction  by  any  number y  midtli)lk\>*  the  fvadum 
by  that  number. 

WRITTEN    EXEHCISEa, 


1.  Multiply  If  by  6. 

PROCESS. 

Or, 
HX6=^«,  =  i^  =  3i 


Analysis. — 6  times  13  twen- 
ty-fourths are  78  twenty-fourths, 
or  3}.    Or, 

Since  dividing  the  denomi- 
nator multiplies  the  fraction 
(Prin.),  6  times  \\  are  ';^,  or  3^. 


RuLR — Multiply  tlie  nutnerator  or  divide  the  denominator  by 
tlie  integer. 


Multiply: 

Multiply: 

Multiply: 

2.  -ft^bvS. 

7.  H  by  7. 

12.  li  by  9. 

3.  ^  by  7. 

8.  ^j  by  6. 

13.  tt  by  13. 

4.  A  by  5. 

9.  A  by  8. 

14.  M  by  14. 

5.  ^  by  3. 

10.  a  by  3. 

15.  -J^  by  18. 

6.  |i  bv  17. 

11.  U  by  7. 

16.  M  by  75. 

MII/l  Il'LKJATION.  Ill 

17.  Wiiai  is  tlx'  value  oi' :i  load  of  17  l)u>lul.s  ol'  a}t[)lr.s  at 
^^^  bik^iel? 

18.  If  a  boy  earua  §§  i)er  day,  how  inueh  can  lie  earn  iu 
9  days? 

19.  At  87  J  a  barrel  what  will  7  barrels  of  flour  cost? 

PROCESS.  Analysis. — In  imihiplying  a  mixed  nunilxjr,  we  niul- 

3  7  f-        tiply  the  fractional  part  and  integer  separately  and  add 
7  the  results. 

~^  Thus,  7  times  $J  =$V  =So}.    7  times  $7  are  $49,  and 

4  9  *        the  sum  of  $49  and  $5^  is  $54}. 

^    -  We  may  reduce  the  mixed  niinibor  to   an  improper 

*^     *        fraction  before  multiplying. 

20.  If  a  man  travel  21f  miles  i^r  day,  how  fur  can  he 
tnivel  in  4  days? 

21.  What  will  13  yards  of  cloth  cost  at  $6J  a  yard? 

22.  If  a  steamship  sails  17J  miles  an  hour,  how  far  can 
she  sail  in  9  hours? 

CASE  II. 

182.  To  multiply  an  integer  by  a  rractiou. 

-1.  Henry  had  0  rabbits  and  sold  ^  of  them  to  James. 
How  many  did  he  sell? 

2.  Jane  had  10  cherries  and  gave  \  of  them  to  her  sister. 
How  many  did  she  give  to  her  sister? 

:].  How  much  is  J  of  818?    ^  of  $18? 

4.  ILjw  nuich  is  -J  of  7  apples?  J  of  9  bu.shcls?  -J-  of  5 
ounces?    •}■  of  3  lemons? 

5.  How  much  is  ^  of  5?    |  of  5?    J  of  5? 
G.  How  much  is  ^  of  7?    |  of  8?    ^  of  12? 

7.  What  is  i  of  36?    |  of  32?    i  of  54? 

8.  How  much  is  |  of  35  tons  ?    ^  of  49  horses  ?    f  of  80? 

183.  Prin'CIPLE. — MuWiphjing  by  a  fraction  is  taking  such  a 
jHirt  of  a  number  as  is  indicated  by  the  fraction. 


11-  COMMON    FRACTIONS. 


WJtlTTEN     EXERCISES. 

1.  Multiply  75  by  f? 

PROCESS.  Analysis.— To  multiply  75  by  f  is 

16  to  find  §  of  75.     §  =  3  tiino*  I.     '  of 

76Xf  =  ^^-^  =  45        ^^  "  ^^»  "»^  1  =  3  timcM  1". 

./^  Or, 

Since  §  =  J  of  3, 1  of  75  =  i  of  3  time#i  75,  or  J^  =  45. 

Rule. — Multiply  ihe  integer  by  the  numerator  of  the  multiplier, 
ami  divide  the  product  by  the  denominator. 
When  ))OMiblc  use  cancellation. 


2. 

Multiply 

9byA. 

8. 

Multiply       51  by  ,s^. 

3. 

Multiply 

17  by  /r. 

9. 

Multiply      79  by  ^. 

4. 

Multiply 

12  by  if. 

10. 

Iklultiply  8318  by  ^. 

5. 

Multiply 

18  by  i*. 

11. 

Multiply  840(5  by  yV 

6. 

Multiply 

100  by  A. 

12. 

.Alultiiil'v  8718  by  ||. 

7. 

Multiply 

144  by  H. 

13. 

Multiply  8825  by  f|. 

14.  A  man  owned  a  mill  worth  87850.     How  much  money 
should  he  receive  for  f  of  it? 

15.  A  s]K)rtsman  shot   48   birds  one  day,  and  ,'•  ;i>  many 
the  next     How  many  did  he  shoot  in  both  days? 

16.  Multiply  4(5  by  5|. 

PROCESS.  Analysis. — In     multii)lyin<j 

4g        Qj,  55.=  V  by  a  mixed  numlxr  we  multi' 

f^s  J.ft  V  *« j  ogg      ply  by  the  integer  and  fraction 

r— -  \2&g       or^i     w^parately  and  add  the  products. 

0 on  J-tAA=  Jo  /  J     Yjjjjg^  ,  ^f  ^Q  jj^  i34^  Q^  27^     5 

^*^^  times  4G  =  230,  and  27-5  +  230 

257f  =257§,  the  product     Or,  we 

may  reduce  tlic  mixed  niiml)er 
to  an  improjier  fraction  and  multiply. 

17.  Multiply  30  by  5} ;     OJ;     7J;     8f?;     10-V 


K 


JklULTlPLICA  1  io.N  1  J  -> 

CASE  III. 
184.  T<»  multiply  a  rrtiction  by  a  fVactioii. 

1.  How  much  is  |  of  4  fifths  of  a  yard?  How  much  is  J 
of  3  ninths  of  a  yartl?     i  of  8  twelfths  of  a  yard? 

2.  If  James  has  f  of  a  dollar,  and  Anna  has  ^  as  much, 
how  much  has  Anna?  If  Henry  has  |  as  much  as  James, 
how  much  has  he?     How  much  is  -J  of  f  ?     |  of  f  ? 

3.  A  man  who  owned  J  of  a  steamship  sold  i  of  his  share. 
AVhat  part  of  the  vessel  did  he  sell?     How  much  is  J  of  |? 

4.  How  much  is  i  of  ft?    1  of  |f  ?    |  of  ff  ?    J  of  fy ? 

5.  If  \  of  a  yard  be  divided  into  2  equal  parts,  what  part 
of  a  yard  will  each  part  be?  How  much  is  .V  of  ^-  yard?  ^ 
of  i  yard? 

G.  At  $5^  a  bushel  what  will  J  of  a  bushel  of  oats  cost? 
How  much  is  i  of  ^?    ^ofJ^?    -J  of  J?    |ofJ?     J  of  "i? 
7.  What  is  the  value  of  ^  of  i?    J  of  i?    |  of  J?    J  of  |? 

WRITTEN   EXERCISES. 

h  Multiply  I  by  |. 

PROCESS.  Analysis. — To  multiply  |  by  f  is  to 

4vA  — i^«— JLi  find  f  of  I,  or  3  times  ^  of  f     iof^  = 

t  A  ■?  —  t  X  5       tt  ^4^^  and  ^  are  3  times  ^U  or  ^ Jf  =  H 

Rule. — Reduce  all  integers  and  mired  numbers  to  %nvproj)er 
fractions. 

MuUiphj  the  niuneralors  together  for  the  numerator  of  Vie 
product,  ami  the  denominators  together  for  Us  denominator. 

,  ^.  When  possible  use  cancellation. 
■J.  The  word  o/"  between  fractions  \a  e<iuiv;ilent  to  tlie  »i'ffn  nfmuUipIi- 
cfttion.    Such  expressions  are  sometimes  called  compound  fnidious.  Thus, 
I  of  X  is  e<iual  to  J  X  i- 

3.  Inte^^ers  may  lx>  expressinl  in  the  f(>rni  of  fractions  by  writing  1 
HA  a  (lenoiniuNior.     Thus,  4  may  be  written  as  ^. 


114                                 <X>MMON    FKACTIOX8. 

Multiply: 

Multiply: 

Multiply: 

2.  A  by*. 

3.  i  by  f . 

4.  A  by  f 

5.  4  by  f 

6.  A  by  A. 

7.  A  by  I 

8.  HbyA. 

9.  II  by  A- 
10.  iJbyA. 

Find  tbe  value  of 

11-  ilxAxAxf 

12.  tfxttxAxA. 

13.  ifxHxHxA. 
14-  Hxifx^xA- 

15.   «XHXf  XA- 

16.  lix-ftxifxA. 

17. 
18. 
19. 
20. 
21. 
22. 

■"•  X  ■"■''  "<   — . 
rX  'X  ::■;  ^:  AV 

i  -  :^  ..  f  xf 
1  X  f  X  A  X  A- 
*  XIX  4  xif. 

Ax«xttxH- 

23.  Multiply  ^  of  1 J  by  ^  of  ^^  of  4. 

rnocEsa.  Analysis.— All  the  mixed 

s  vy  7  vy  5  vy   9   X  4'^  =  5-   Ot  W       numbers  must  be  eliange<l  to 

improiKT  fractions,  and  all 
whole  numbers  expressed  in  the  fractional  form.  Multiplying  and 
caiwdliny  we  have  ^,  or  1  J. 

24.  Multiply  ^  of  i  of  o  by  ^  of  f  of  U. 

25.  Multiply  |  of  ^  of  8  by  f  of  ^^^  of  15. 

26.  Multiply  3J  times  4  by  4  times  J  of  7. 

27.  Multiply  5}  times  J  of  18  by  f  of  3  times  \  of  4. 

28.  A\Tiat  will  be  the  cost  of  ^^  of  a  yard  of  cloth  at  8f  a 
yard? 

29.  16J  feet  make  a  rod.     How  many  feet  are  tliere  in  5^ 
rods? 

30.  A  man  who  owned  f  of  a  mill,  sold  j-  of  his  share. 
AVhat  part  of  the  mill  did  he  sell? 

31.  How  many  yards  of  cloth  are  there  in  12^  pieces  of 
cloth,  each  piece  containing  42|  yards? 

32.  At  81 6 J  per  ton  how  nuicli  can  be  realized  from  the 
sale  of  4^  tons  of  hay  ? 


DIVISION.  115 

3;;.   Wluit  is  the  value  of  |  of  -^  of  fj  of  ^  of  15? 
34.  AVhat  is  the  value  of  -^  of  ^  of  ff  of  29? 


DIVISION. 

CASE  I. 
185.  To  diviflc  a  fVacfion  by  uii  integer. 

1.  Mr.  Allen  divided  3  fourths  of  a  dollar  equally  between 
■*»  l)oys.     How  much  did  each  receive? 

2.  A  man  divided  f  of  an  acre  into  3  equal  lots.  How 
large  were  they? 

3.  If  7  yards  of  cloth  were  bought  for  ^,  what  was  the 
cost  per  yard? 

4.  If  4  books  cost  $^,  what  is  the  cost  of  each? 

*  5.  Mr.  Hurd  put  f  of  his  crop  of  wheat  on  3  wagons. 
What  part  of  his  crop  was  on  each  wagon?  How  much 
i8i-^3? 

6.  In  divi<H?ig  a  fraction,  what  part  of  the  fraction  is 
divided .' 

7.  A  gentleiiiiin  divided  V  a  barrel  of  flour  equally  between 
2  people.  What  i)art  of  a  barrel  did  each  receive?  How 
much  is  ^-7-2? 

8.  3  boys  spent  altogether  3J.  If  eacli  spent  the  same 
amount,  what  jxirt  of  a  dollar  did  each  spend?  How  much 
is  J-f-3? 

9.  Mr.  Smith  divided  \  of  his  farm  into  3  equal  fields. 
What  part  of  his  farm  did  each  field  contain  ?  How  much 
is  J  -T-  3?     How  is  the  result  obtained  from  the  fniction  ^? 

10.  In  what  other  way,  then,  liesidcs  dividiiiL^  tlic  mmiora- 
tor  may  a  fraction  l)e  divided  ? 

11.  When  a  numl)er  is  dividt'd  by  ;>  what  pan  «;i  w  is  i.umd? 

12.  When  a  fiiution  is  divided  by  7  what  jwirt  of  it  is  found? 


ik; 


CX)MMON    FRACTIONS. 


13.  Whatis^of^?    f-v-3?    iofW?    A-^^? 

14.  What  is  the  value  of  I H- 2?    Of|-f-3?    Of  ^ 


9? 


-  18fi.  Principle. — Dividing  tiie  numerator  or  multij)lijing  the 
denominator  of  a  fraction  by  any  number  ^  divides  the  fraction  by 
dial  number. 


wnrrrry   k  y  /  /.  /  i  s  rs. 


1.  Divide  fj  by  (i. 

PROCiSS. 

Or, 


Analysis. — Since  dividing  the  nu- 
merator of  a  fraction  divides  the  frac- 
tion, the  fraction  \}  may  be  divided  by, 
C  by  dividing  the  numerator  by  6.  Or, 
Since  muhiplying  the  denominator 
divides  the  fraction,  the  fraction  may 
be  divided  by  6  by  multiplying  the  denominator  by  6.  The  result  by 
botli  processes  b  ^. 


^     Rule. — Divide  iJie  numerator  or  multiply  Hie  denominator  by 
€ie  given  integer. 


Divide: 

2.  f  by  4. 

3.  4  by  8. 

4.  U  by  3. 

5.  H  by  6. 


Divide : 

6.  ^  by  8. 

7.  44  by  15. 

8.  ki  bv  7. 

9.  li  by  18. 


Divide : 

10.  fj  by  21. 

11.  a  by  18. 

12.  15  by  38. 

13.  Mi>y35. 


14.  Divide  16f  by  5. 

PROCESS. 

16f==¥ 

Or,     5)16f 
3^ 


Analysis. — We   may  reduce  ICf  to 

an    improjjer  fraction,  and    divide    as 
before.    Or, 

We  may  divide  without  reducing  to 
an  improj)cr  fraction.  Thus,  5  is  con- 
tained in  ICJ,  3  times  and  a  remainder 
of  1^,  or  I,  and  |  divided  by  5,  equals 
^.     Therefore  the  quotient  is  S^. 


1)1\  1>1<).\. 

1 

Divide : 

Divide : 

17.  38f    by  8. 

18.  24/t  by  9. 

19.  S^  by  10 

20.  25f  by  7. 

117 

Divide : 

15.  17J  by  6. 

16.  25f  by  4. 

_M.  A  man  gave  each  of  his  5  sons  an  equal  share  of  ■} 
of  liis  estate.     What  part  of  the  whole  did  each  receive? 

22.  8  men  built  J  of  a  mile  of  wall  in  10  days.  AVhat 
part  of  a  mile  did  each  man  build  daily? 

23.  Mr.  B.  earned  835 J^  by  working  8  days.  How  much 
did  he  earn  per  day? 

24.  I  \md  eiOJ^  for  27  pounds  of  butter.  What  did  I  pay 
a  ix)und? 

25.  A  farmer  realized  $233J  for  21  bushels  of  clover  seed. 
How  much  did  he  get  per  bushel? 

26.  Three  men  who  have  been  partners  in  business  gain 
$3216}.  If  they  share  equally,  what  will  be  each  one's  part 
of  "the  gain? 

CASE  II. 
187.    To  divide  an  integer  hy  a  fraetioii. 

1.  How  many  fourths  are  there  in  an  apple?    In  2  apples? 

2.  How  many  apples,  at  4  cent  each,  can  be  bought  for  2 
cents?     For  3  cents?    For  10  cents? 

3.  At  $J  a  yard,  how  many  yards  of  cloth  can  be  bought 
for  81?    For  82?    For  83?    For  810? 

4.  At  $\  an  ounce,  how  many  ounces  of  nutmegs  can  be 
l)ought  for  82?  How  many  at  8 J  an  ounce  can  be  bought 
for  82? 

5.  If  a  man  can  mow  4-  of  a  field  jjer  day,  how  long  will 
it  take  him  to  mow  the  entire  field?  How  long  if  he  can 
mow  ^  per  day  ? 

6.  At  8^  a  bushel,  how  many  bushels  of  lime  can  be  bought 
for  W?  At  8f  a  bushel  how  many  bushels  cm  1><'  Unight 
for  $5? 


118 


CXJMMON    FRACTIONS. 


7.  AVhen  apples  are  worth  |  of  a  cent  apiece,  how  many 
can  be  bought  for  12  cents?  How  many  times  is  }  contained 
m  12? 

8.  How  many  pieces  of  cloth  J  of  a  yard  long  can  l)c  cut 
from  a  piece  9  yards  long?  How  many  times  is  f  contained 
in  9? 

9.  How  many  times  is  f  contained  in  8?    f  in  9? 

10.  AVhat  is  the  quotient  when  8  is  divided  by  |?    10  by  f  ? 

11.  AVhat  is  the  value  of  10 --I?    7---|?    9^f? 


WniTTBN   BXMBCI8X8. 


1.  Divide  12  by  f 


raocEss. 

2 


12-f-^=-ilfL=14 

Or, 
12=i^;i^-^^=14 


Analysis. —  |  is  contained  in  12 
7  times  12,  or  84  times;  and  ^,  one- 
sixth  of  84  times,  or  14  limes.    Or, 

Reducing  12  to  sevenths,  we  have 
84  sevenths.  6  seventlis  are  contained 
in  84  sevenths  14  times. 


y    Rule. — Multiply  Hie  integer  by  Vie  denominator  of  the  fraction 
and  divide  Uie  product  by  the  numerator.  ^  Or^ 

Reduce  the  dividend  and  the  divisor  to  similar  fractiom,  and 
divide  the  numerator  of  Hue  dividend  by  Hie  numerator  of  the 
divisor. 


ivide : 

Divide : 

Divide: 

2.  18  by  f 

10.  72  by  f 

18.  51  by  U. 

3.  64  by  I 

11.  34  by  f 

19.  69  by  If. 

4.  26  by  i. 

12.  15  bv  «. 

20.  70  by  ^. 

5.  48  by  ^. 

13.  49  by  H. 

21.  65  by  i^. 

6.  75  by  H. 

14.  91  by  y. 

22.  90  by  Jf . 

7.  Slby^V 

,      15.  64  by  if. 

23.  36  bv  H. 

8.  45  by  U. 

16.  54  bv  ^. 

24.  39  by  M. 

9.  39  by  tV 

17.  39  by  if. 

25.  24byH. 

DIVISION. 


119 


2(1.  Divide  24  by  3^. 


PROCESS. 


L'  } 


Or 


3} 

7 


24 

48^ 
6f 


Analysis. — When  the  divisor  is  a 
mixed  number  we  reduce  it  to  an  im- 
proper fraction  and  proceed  according 
to  the  rule.  Thus,  3J  =  |;  and  24  di- 
vided by  i  =^  48,  and  by  |  =  f  of  48, 
or  6^.    Or, 

We  may  reduce  3.^  to  halves,  and 
24  to  halves,  and  divide  the  numera- 
tor of  the  dividend  by  the  numerator 
of  the  divisor. 


Divide  the  following  by  Iwth  processes : 


27.  15  by  3i. 

28.  23  by  6f. 


29.  26  by  7f . 

30.  35  by  6f 


31.  39  by  8f 

32.  46  by  7^^. 


33.  At  $1  a  yard,  how  many  yards  of  cloth  can  be  bought 
for  $15? 

34.  When  com  is  worth  $J  a  bushel,  how  many  bushels 
must  a  man  sell  to  get  money  enough  to  pay  §18  taxes? 

35.  A  man  invested  $32  in  i^eaches  at  §lf  ix^r  basket. 
How  many  Iwiskets  did  he  purchase? 

36.  AVhat  is  the  price  of  hay,  when  34  tons  sell  for  837? 

37.  Mr.    Shaw   jwiid   $6  for   14|   i)ounds  of  Java  coffee. 
What  did  it  cost  him  jx^r  jwund? 

38.  It  requires  656  jwunds  of  meai  lo  .supply  oi  boldiers 
3^  weeks.     How  much  does  each  soldier  eat  daily? 


CASE  III. 
188.  T<»  <livi<ie  a  fVaetion  by  a  fVaotioii. 

1.  How  many  fourths  are  there  in  1?     How  many  fifths? 
eighths?     tenths?    fifteenths?    twentieths? 

2.  How  nuiny  pieces  of  cloth  \  <  id  long,  lau   1h' 
cut  from  a  yard? 


1  _"  '  ( »^I  ^l'  -N     1  i;  \«   I  K  >.\-. 

.'».    If"  il  •  t   \  :i(]   l()n:,%  how  luaiiy  would 

tlu  Tr    1  :il|)iire   Uitll    tlir    lllllnlilT 

wlicu  (Iju  |>iccc'6  ai(j  -J  ul' a  yard  ]»»ii_  .' 

I.    lf*~llu'  nieces  wrrc  f  of  a  yard    loij^%  liuw  liiuiiy  would 

ilii  iiiiihIk  I  (I  pieces  comjMire  with  the 

iiuiiilu  1   wliiu  ihe  ])iL'C'c'.s  arc  ^  <.t'a  yard  long? 

:..    ll..\v  many  timi'S  is  1  (..maincd  in  1?    |?    |?    J?     |? 

<■).  SiiKT  [  i-«  coiiiaiiicil  in  I,  ri-lit  times,  how  many  times 
will  it  Ix!  ccnlaiiird  in  V!  What  pait  cf^  times  will  it  Ik? 
contained  in  I  / 

7.  Since  .;•  i-  .'Miiaiiicd  in  1  -J  of  8  time-.  (»r  f  times,  how 
many  time-  will  ii  he  eoiiiaiiicd  in  J?    How  many  times  in  -J? 


S.    What   i-  th.    value  of  1-T-l?     Of  l--f  ?     ^-4-^?  \-^1 
*J.   Into  how  many  parts  of  H  eiLdiths  of  a  dollar  each,  can 
6  eighths  of  a  dollar  be  divided  / 

10.  IIow  many  sacks  containing-  j',,  nf' a  harrel  <  ach.  can  be 
filled  from  y'^  of  a  barrel  nf  tlnur?  Ilnw  many  times  is  ^ 
contained  in  -j^?     ?  in  5  ':*       ■,  in   .".,  ?      ,'-  in  //? 

1 1.  llnw  many  pine-api  cacli.  can  !>*■  I).Mii:ht  i"nr  8.^  ? 

12.  liow  many  times  i-  ;  c..iuained  in  Y-     »  J"  1  •'     s  '"  4  '• 

13.  How  many  limes  is   ),   contained  in  1?     In  Vi     In  Vi 

14.  lL>w   many  times  is   z   eonlained  in  1?     In  i?     I"  i '■^ 


wniTTrx   rx  rn ciSES. 

1.  Divide  ^  ^  - 

TK'     .^  -  Analysis. — i    is  contained  in  1,  5 

4_^3.__4  y^   5  :^—  2^  times;   and   §  is  contained  in  1,  one- 

°  third  of  o  tir>i<^«   rn-  '  tinie.«i. 

(hy  And   >in^  iiained    in   1,  ^j 

4    .    ^ 20_i_'>i-— '>o  times,  in   1   it  \viii    lu-  contained  ^-  of 

T"^^  —  -Jo-^to— tT  |=:|o  tin^t.^_     Or. 

^  is  equal  to  §2,  and  f  is  equal  to  |i.     21  thirty-fifths  are  contained 
in  liu  ihirty-tiftlis  |£  times. 


IHVISION. 


121 


Y  Rule. —  Mttltiplt/  the  divideml  by  the  diri-^or  inverted.,  Or, 
Jiednce  the  dividend  and  divisor  to  similar  fractions  and  divide 
die  nunieraior  of  Hie  dividend  by  tlie  numeraior  of  Hie  divisw. 

X  When  possible  use  cancellation. 


ivide: 

Divide : 

Divide : 

2.  iJ  by  J. 

3.  HbyH. 

4.  ^IbyT-V 

6.  HbyxV 

7-  H  by  f. 
8.-^byA. 
9.  If  by  A. 

10-.  H  by  A 

11.  Hby-B 

12.  H  by  A 

13.  f^byi^r 

14.  What  is  the  quotient  of  |  of  J  of  5f  divided  by  f  of  ^ 

ofj? 

PROCE&s.  Analysis. — In  the  bo- 

I  of  f  of  V-  -^  f  of  f  of  J  =  ^"**o"   ^^  examples  like 

J  this,  all  mixed  numbers 

-|X4X  VX|XiXf=V=-^^    Bhoukl  be  cliangcd  to  im- 

proi)er  fractions,  and  all 
fractions  that  are  factors  of  the  divisor,  inverted,  and  the  product  found 
as  in  previous  examples. 

15.  Divide  f  of  f  of  16  by  f  of  f  of  5J. 

16.  Divide  ^  of  |f  of  5 J  by  4 J  times  ^  of  16. 

17.  Divide  J  of  J  of  f  by  |  of  -J. 

18.  Divide  i  of  |  of  ^V  by  5  times  f  of  f 
ly.  Divide  |  of  f  of  15  by  f  of  ^  of  6. 

20.  Divide  f  of  ^V  of  22  by  ^  of  ^  of  16. 

21.  Divide  j\  of  3 J  of  6  by  |  of  6  times  If 

22.  Divide  H}  times  J  of  7'by  f  of  |  of  5. 

23.  How  many  pieces  of  ribbon  yV  of  a  yard  in  length, 
can  be  made  from  J  of  -fjf  of  a  yard  ? 

24.  If  a  man  sjxjnds  $J  i)er  day  for  cifrar?,  in  how  many 
days  will  he  sjiend  31 7^? 

25.  How  many  yi\v(]<  nf  clotli   at  t>oj  j,)cr   vanl  can   be 
bDught  for  631 7J? 


122  COMMON   FRACTIONS. 

26.  At  $1  per  biH^"!    l-"w  many  1)ii>1h1>  <>[  |»,uii..v>  ran 
Ixj  Iwugbt  for  $17  J  y 

27.  If  a  family  us«  r     .i  a  barrel  of  flour  a  week,  how  long 
will  5}  barrels  la.st? 

28.  If  a  \yoy  earns  S^  daily,  how  long  wOl  it  take  him  to 
earn  83 j? 

29.  A  certain    nninlKi-    iiitilii|.rK(l  by   2  i«    equal    to    -.1. 
AVhat  is  that  number? 

30.  If  a  man  can  saw  1\  conls  of  wqo<1  in  one  day,  how 
long  will  he  require  to  saw  17^  cords? 

31.  If  a  horse  eats  12^.  bushels  of  oats  in  5  weeks,  how 
much  does  he  eat  in  a  day? 

32.  When  wheat  is  selling  at  81J  per  tush^l    1  <  \^   Mi:iny 
bushels  can  be  bought  for  83168? 


FRACTIONAL  FORIMS. 

189,  Expressions  of  unrxocMtcd  division  arc  ofton  uritim 
in  the /onu  of  a  fraction, 

100.  A  fractional  form  liaving  an  integral  denominator 
and  a  fractional  numerator  is  called  a  Comjjlex  Frac- 
tion. 

Thus,  -|  and  -jT  are  complex  fractions. 

Expressions  which  have  a  fraction  in  the  denominator  can  not 
properly  be  regarded  as  Complex  Fractions,  though  they  are  commonly 
classified  as  such. 

1.  Find  the  value  of  the  fractional  form  J. 

I 

PROCESS.  Analysis. — ?-   is    an   expression 

*=4-i-X==^X'f=#T     f>f  division,  and  is  the  same  as  ^  -h- 1, 
"8"  3  wliich  is  equal  to  \^. 


IKALTIONAI.    lii;i 

\  1  1  (  1 \     (  )  I'     \  I 

MMEllS.              123 

[rdllcr    I"   S 

iraple  fraction . 

6.1. 
6f 

10.  'A. 

6i 

14.  ^  of  «. 
1  of  9 

:! 

7.1. 

11.  tl. 

15.  *  of  3 
6 

4.^. 

() 

1 

12.  if. 

16.       »      .• 
iofj 

5.  it 

13.  Li. 
U 

17.  *  of  i 
4iX3 

FRACTIONAL  RELATION  OF  NUMBERS. 

CASE  I. 
191.  To  fiud  the  relation  of  one  number  to  another. 

1.  AVhat  part  of  5  cents  is  1  cent?    2  cents?     3  cents? 
4  cents? 

2.  What  part  of  0  acre?  \>  5  noro.v?    7  arrr,=:?    8  aero.'??     4 
acres? 

^    3.  What  part  of  4  apples  is  1  apple?     ^  of  1  appk  .^      ] 
of  1  apple?     ^  of  1  apple? 

4.  What  part  of  35  is  ?2?     W    8^?     $i?    ^2 

5.  AVhat  partof  86  is  SI?     W     H-     H^ 

6.  Henry  had  So  and  gave  his  brother  S|.     What  part  of 
his  money  did  he  give  his  brother? 

7.  Jajnes  earned  S7,  :m<l  ]\\<  liroflxT  82.     What   jiart  of 
the  whole  did  each  car 

Principle. — Only  like  numbers  can  have  relation  to  eaclt  oUier. 

8.  What  is  the  relation  of  5  to  9? 

Analysis.— 1  is  I  of  it.  and  .'»  !■<  5  tiiiK»»  J  ,.- 
Honcv  f)  is  I  of  U. 


124  CX)MMON   FB ACTIONS. 

What  is  the  rektion 


9. 

Of   7  to  21? 

12. 

Of   9  to  18? 

15. 

Of  15  to  24? 

10. 

Of  12  to  16? 

13. 

Of   8  to  32? 

16. 

Of  14  to  35? 

n. 

OnO  tn  'JH? 

n. 

()r'^l>  to  48? 

17. 

Of  18  to  -^1  ? 

la.  What  is  tlie  relation  of  f  to  2  ? 

Analysis.— 1  is  J  of  2.  and  f  of  1  ia  f  of  i  of  2,  or  ^q  of  2. 
Hence  |  is  -^^  of  2. 


What  is  the  relation 


19.  .Of  f  to  4? 

20.  Of  \  to  9? 

21.  Of  f  to  6? 


22.  Of  J  to    8? 

23.  Of  I  to  15? 

24.  Of  1^  to  25? 


25.  Of  I 

26.  Of  I  to  18? 

27.  Of  J  to  12? 


28.  What  is  the  relation  of  J  to  f? 

Analysis,— f  b  |  of  f ,  and  1  is  7  times  J  of  2, 
1  is  I  of  f,  I  of  1  is  I  of  I  of  ^,  or  H  of  f 
Hence  |  is  \\  of  f . 


AVhat  is  the  relation 


29.  Of  J   to  I? 

30.  Of  I   to  j? 

31.  Of -ft-  to  4? 


32.  Of  4  to  T^? 

33.  Of  4  to   I  ? 

34.  Of  4  to  ^  ? 


35.  Of  I   to  f  ? 

36.  Of  ^   to  I? 

37.  Of  -A  to  I? 


CASE   II. 

192.  A  uumber  and  its  relation  to  another  number 
given,  to  find  the  other  nnniber. 

1.  2  cents  are  J-  of  how  many  cents?     ^  of  how  many  cents? 
\  of  how  many  cents? 

2.  3  is  ^  of  what  number?     \  of  what  number?    \-  of  what 
number? 

3.  8  is  ^  of  what  number?     f  of  what  number?     f  of  what 
number? 


KiAii.w    i;\i:ia:i.si:s.  12o 

4.  12  is  i"  of  what  immht'i?     3  of  wliat  nuinl)cr?     I  of 
"what  number? 

5.  i  is  J  of  what  number?     \  of  what  numl)er? 

6.  "i  is  i  of  what  number?     *  of  what  number? 

7.  24  is  f  of  what  number? 

Analysis, — Since  24  is  f  of  a  certain  number,  1  fifth  of  the  num- 
ber is  \  of  24,  or  6;  and  since  0  is  ^  of  the  ninnber,  the  nninlxr  must 
be  5  times  6,  or  30.    Hence  24  is  ^  of  30. 

8.  24  is  ^  of  what  numl)er?     56  is  J  of  what  number? 

9.  18  is  5  of  what  number?     49  is  ^  of  what  number? 

10.  24  is  ^  of  what  number?     42  is  f  of  what  number? 

11.  45  is  f  of  what  number?    96  is  j-|  of  what  number? 

12.  ^  is  J  of  what  number? 

Analysis. — Since  ^  is  |  of  a  certain  number,  J  of  the  number  is  \ 
of  ^,  or  2 ;  and  since  ^  of  the  number  is  S,  the  number  must  be  3  times 
f ,  or  J.    Hence  J  is  3  of  £. 

13.  -j-f  is  f  of  what  numl^er?  |f  is  y^  of  wliat  numl)er? 

14.  J^  is  ^  of  what  numl)er?  -fi}  ^®  V  of  what  numl)er? 

15.  ^  is  ^Tf  of  what  numl)er?  J-J  is  ^  of  what  number? 

16.  if  is  T^  of  what  number?  |-J  is  -^^  of  what  number? 


REVIEW  EXERCISES. 

103.  1.  ^fr.  B.  liought  a  barrel  of  flour  for  S7J,  a  cord  of 
wood  for  ^i)},  and  gave  the  clerk  a  twenty-dollar  bill.  IIow 
much  change  should  Mr.  B.  receive  ? 

2.  A  merchant  lK)ught  360  jwunds  of  sugar  at  11^  cents 
a  jwund,  50  jxiunds  of  tea  at  62!  cents  a  pound.  How  much 
did  he  jMiy  for  both  ? 

3.  If  a  man  can  cut  in  one  day  .1  of  a  Held  eontainiiig  7 
acres  of  wheut,  how  many  acres  can  he  cut  in  f  of  a  day  ? 


126  CX)MMOX    FRACTIONS. 

4.  What  will  be  the  cost  of  3j\  dozen  eggs  at  18}  cents  a 
dozen? 

5.  If  a  man  can  hoe  a  iitld  in  7]  days,  how  long  will  it 
take  3  men  to  hoe  a  field  2 J  times  as  large? 

6.  Fn)m  a  Iwirrel  of  kerosene  containing  4H  gallons  it  was 
estimatcnl  that  I  lcake<l  out.  If  I  paid  §6.15  for  it,  at  what 
price  per  gidlon  must  I  sell  the  remainder  to  balance  the  loss 
8ustaine<l  by  leakage? 

7.  James  HendcrH»n  >^M  ;  of  his  farm  of  155  acres  to 
Mr.  Paine,  and  Mr.  Taint  .-omi  x.jd  J  of  what  he  had  lK)ught, 
to  Mr.  Banker.     How  many  acres  did  Mr,  Banker  buy  ? 

8.  Mr.  A.  built  a  block  of  stores  which  cost  him  §3122} 
for  brick,  813081  for  lumber,  $3258f  for  lalK)r,  and  §1325iV 
for  other  expenses.  He  sold  the  block  for  810000.  Did  he 
gain  or  lose,  and  how  much  ? 

9.  There  are  272J  8(]uare  fc(  i  in  a  -juai'  vm].  How 
many  square  feet  are  there  in  f  of  a  square  ro<l 

10.  I  sold  a  house  and  lot  for  83215,  which  \mi  s 
what  it  cost  me.     How  much  did  it  cost? 

11.  How  much  will  8  carpenters  earn  in  6|  days  a 
per  day? 

12.  If  a  man  walks  3 J  mile?  per  hour,  in  how  many  hours 
can  he  walk  30 J  miles? 

13.  Mr.  Jones  left  an  iMai*.-  \aiimi  m  ..  i.m*ww.  .  ^,i  it 
was  divided  equally  among  4  sons  and  the  rest  equally  among 
3  daughters.     What  was  the  share  of  each? 

14.  The  price  of  maple  sugar  this  year  i?  only  l  ui"  w  luit  it 
was  last  year.  How  much  more  would  I  have  received  last 
year  for  3140  jwunds  which  I  sold  this  year  at  8 .20  a  pound? 

15.  Custom  millei-s  take  ^  of  tlie  quantity  of  grain  as  pay  for 
grinding  it.  How  many  bushels  must  a  man  carry  to  the  mill 
so  tliat  he  may  bring  back  14  bushels  of  ground  provender? 

16.  If  845  IS  f  of  my  money,  wlmt  part  of  it  will  that  sum 
plus84Jbe? 


RKVIEU    EXERCISES.  127 

17.  A  iUrmer  had  two  fields  in  which  he  kept  his  sheep. 
In  one  there  was  ^  of  the  whole  number  of  sheep,  and  in  the 
other  there  were  148  sheep.     How  m:iny  sheep  had  he? 

18.  A  merchant  exchanged  21  barrels  of  flour  worth  $7f  a 
barrel,  for  24 J  cords  of  wood.  What  did  the  wood  cost  him 
ixn*  cord.?  » 

19.  If  I  give  A  J  of  my  money,  B  -}  of  it,  and  C  J  of  it, 
what  part  of  my  money  have  I  left? 

20.  Mr.  A.  owns  f  of  a  vessel  valued  at  318326.  If  he 
sells  f  of  his  share  to  Mr.  B.,  what  part  of  the  whole  will 
he  have  left?  What  part  will  Mr.  B.  have?  What  is  the 
value  of  Mr.  A.'s  share?     Of  Mr.  B.'s  share? 

21.  Alter  buying  a  suit  of  clothes  for  $60  I  found  I  had  f 
of  my  money  left.     How  much  had  I  at  first? 

22.  A  man  sold  .}  of  31  cords  of  wood  for  f  of  $8^.  How 
much  did  he  receive  for  it  per  cord? 

.23.  How  many  tons  of  hay  will  be   required  to  keep   7 
horses  for  6  months,  if  0  horses  eat  16 J-  tons  in  that  time? 

24.  If  J  of  a  farm  is  sold  for  88516,  what  would  be  the 
worth  of  the  whole  at  the  same  rate  ? 

26.  A  gentleman  spent  ^  of  his  annual  income  traveling, 
and  ^  of  the  remainder  in  the  purchase  of  books.  The  rest, 
which  was  88526,  he  exjKinded  upon  paintiiiirs  and  other 
works  of  art.     What  was  his  annual  income 

20.  Two  men  dug  a  ditch  for  853;  one  inuii  >>wiktd  oj^ 
days  and  dug  14J^  rods;  the  other  worked  as  many  days  as 
tlie  first  dug  rods  j^er  day.  How  much  did  each  receive,  if 
they  shared  in  projxjrtion  to  the  time  they  worked? 

27.  Two  brothers  together  own  ^  of  a  flouring  mill  valued 
at  813000.  One  owns  J  as  much  as  the  other.  What  is 
the  value  of  each  one's  share? 

28.  The  I0.SS  caused  by  a  fire  was  83865.  The  sum  was 
I>aid  by  an  insurance  company  which  insured  the  stock  for  J 
of  ita  value.     What  was  the  entire  value  of  the  stock? 


128  COMMON    FRACTIONS. 

29.  A  can  do  a  piece  of  work  in  10  days.  What  jMirt  of  it 
can  hv  <1m  in  1  day?  If  B  can  do  the  same  piece  of  work  in 
8  days,  wltai  jMirt  of  it  can  he  do  in  1  day?  What  jMut  can 
both  tonrethor  do  in  1  day?     How  many  day?  would  ]k»  re- 

(liiir.  <1  f''i   '  lo  the  work? 

•■;i>.  HA  .  .Ill  Mw  u  piece  of  work  lu  .>  iiaw--  juki  ii  m  --  uavH, 
lu.w  1'  11-  will  it  take  l)oth  to  do  it? 

31.  If  A  and  Ji  can  do  a  piecxi  of  work  in  10  days,  and  A 
can  do  it  alone  in  15  days,  liow  long  will  it  take  B  to  do  it? 

IV2.  A  tire  124  feet  high  was  broken  in  two  pieces  by  falling. 
f  of  the  length  of  the  shorter  piece,  eqnaled  f  of  the  length 
of  iho  longer  piece.     What  was  the  length  of  each  piece? 

•M.  A  nmn  who  had  sjxjnt  ^  his  money  and  $J  more,  fonnd 
that  he  had  $21  left.     How  much  money  had  lie  at  fir>t  ? 

^^^     ^    B  and  C  can  do  a  j)iecc  of  \\ork  in  9  chiys.     A  tan 
)  days,  and  B  in  30  days.     In  what  time  can  C 
do  it? 

^i>.  Two  laiUiers  will  togct!.^ .  j... .  ivach  the  top  of  a  build- 
in::  7")  feet  high.  If  the  shorter  ladder  is  ^{  the  length  of  the 
longer  one,  what  i<  the  length  of  each? 

36.  There  UiT  '^*"  '^nnbers  whose  sum  i-  M<>  <""•  ..rAvl.i.-h 
is  f  the  other.  n-e  the  numbers? 

'>7.  In  1S75  a  merchant's  profits  were  4-  of  his  receipts;  in 
187i''  ill*  V  'hM'n'Mv.i]  1,  which  diminishr-*]  his  unints  I  of 
827  •;.     Wh:  i<  receipts  in  187."> 

38.  A  and  B  together  had  85700.  |  of  As  iiioiu y  \\;i- 
equnl  to  ^  (^f  B's.     How  much  had  each? 

in  engaged  to  work  a  year  for  8240  and  a  suit 
of  clothes.  At  the  end  of  9  months  an  equitable  settlement 
was  made  bv  iriving  him  S168  and  tlie  suit  of  clothes.  What 
was  the  value  (if  ihe  clothes? 

40.  A  and  B  can  do  a  piece  of  work  in  12  days.  Assum- 
ing that  A  can  do  -J  as  nnich  as  B,  how  long  will  it  take  each 
to  tlo  it? 


194,  1.  If  an  apple  be  divided  into  ten  equal  parts,  what 
part  of  the  apple  will  one  of  these  parts  be?  Two  parts? 
Five  parts? 

2.  If  one-tenth  of  an  apple  l)c  divided  into  ten  equal  parts, 
what  part  of  the  apple  will  one  of  these  parts  be?  Two  parts? 
What  part  of  1  is  ^\  of  ^\  ?    yV  "f  tV  ?    VV  of  tV- 

3.  If  one-hundredth  of  u  dollar  be  divided  into  ten  equal 
jyarts,  what  part  of  a  dollar  will  oneof  thcst^  parts  be?.  Eight 
jMirts?    AVhat  part  of  1  is  ^  of  y^y ': 

4.  What  jmrt  of  one-tenth  is  one-hundredth?  Of  2  tenths 
is  2  hundredths?  Of  ?>  tenths  is  8  hundredths?  Of  9  tenths 
is  i)  hundredths? 

f).  What  part  of  onc-liuiidivdlli  is  oiit'-tliousandth?  Of  2 
hundredths  is  2  thousandths?  Of  8  hundredths  is  8  thou- 
^;uldths? 

<).  What  are  the  divisons  of  any  thing  into  tenths,  huu- 
ilicdths,  thousandths,  ten-thousandths,  etc.,  calle<l? 

Ans.  Decimal  divisions. 

DEFINITIONS 

195.  A  Deeitnal  Fraction  is  one  or  moro  of  the 
df.'cimul  divisions  of  a  unit. 

The  wonl  decimal  is  tlerivcd  fn»in   llu    l.aim  word  dccnn, 

which  sijrnifies  ien. 

9  (»'-"■» 


130 


DECIMAL   FRACTIONS. 


Decimal  fractions,  ibr  the  sake  of  brevity,  are  usually  called 
decinuUs. 

VM.  Since  tenths  arc  equal  to  ten  times  as  many  hun- 
dredths, and  hundredths  are  equal  to  ten  times  as  many 
thousandths,  thousandths  to  ten  times  as  many  ten-thou- 
sandths, etc.,  it  is  evident  that  decimals  have  the  same  law 
of  increase  and  decrease  as  integers,  and  that  the  rfe;jo?  ■ '"''^r 
may  therefore  be  indicated  by  the  position  of  Uie  figun 

According  to  the  decimal  system  of  notation,  figures  de- 
crease in  tenfold  ratio  in  passing  from  left  to  right ;  therefore 
a  figure  at  Vie  right  ofmiiOi  will  express  teniJuf,  at  the  right  of 
tentJis,  huiidredthSy  at  tlie  right  of  hxindredUiHy  tJiouaatidtJiSy  etc., 
as  is  exliil)ittHl  l>v  tl»..  f^,ll,.\viii 


i>v  ikrr><vii 


234 
147 
313 
164 
164 


5  =234,^ 

0  6  =  147t^ 

005  =313y^ 
0  3  4  =  164^^7 
03400=164,^^ 


0164 


■Uh 


From  this  mode  of  expressing  decimal  fractions  the  follow- 


ing principles  are  deduced; 


197.  Principles. — 1.  Decimcds  conform  to  the  same  prin- 
ciples  of  notation  as  integers. 

2.  Each  decimal  cipfier  prefixed  to  a  decimal  diminisJies  its 
value  tenfold,  since  it  removes  eacJi  figure  one  place  to  ilie  right. 

3.  Annexing  ciphers  to  a  decimal  does  not  alter  its  value,  since 
ii  does  not  change  the  place  of  any  figure  of  the  decimal. 

4.  The  denominator  of  a  decimal,  wlien  expressed,  is  1  imih  as 
many  ciphers  annexed  as  there  are  orders  in  the  dedmcd. 


Tn:("nrAT>  fractions.  131 

The  Tfecimal  l^oint  is  a  period  placed  before  the 
dociinal.     Thus,  .0  represents  -{'q;  .54  re[)resents  -{^fj. 

The  decimal  point  is  also  called  the  SrparatriXy  since  it  is 
also  used  to  separate  integei*s  from  decimals. 

198.  A  l^ure  Uechnal  Number  is  one  which  con- 
sists of  decimals  only;  as  .387. 

199.  A  Mixed  Decimal  Ntnnher  is  one  which  con- 
sists of  an  integer  and  a  decimal;  as  46.3,  which  is  equal  to 
40^. 

2(K).  A  ConiJ>lex  Decimal  is  one  which  has  a  com- 
mon fraction  at  the  right  of  the  decimal;   as  .3J,  which  is 

ecjual  to     t. 

*  10 

NUMERATION  TABLE. 


0^ 


.2   '^   "i:    Si'  '^      .    «:       r:=   ^    Si'   •£  n3   .2    c   'T3    5  ^ 
-'     7     '.     4    8    3     4    .    (')    8    4    7    3    0    2    4    L'     :. 


Intecjers.  Decimals. 

By  examining  this  table  it  will  be  seen  that  ienOis  occupy 
the  jiM  decimal  place,  hundredths  the  second,  thousandths  the 
Oiird,  ten-thousandths  the  fourth,  hinuhrd-fh()ii.«i)idthf<  the  fifth, 
millionths  the  sixUiy  etc.     Hencv, 

The  place  occupied  bt/  any  ordrr  of  dtcimals  is  one  /'. •<.««•  than 
that  occupird  by  the  cvrresponding  order  of  integers. 


132 


DECIMAL   FRACTIONS. 


tiOl.  What  onler  of  decimals  occupies 


Ist  place? 
4th  place? 
3il  place? 
7th  place? 


5th  place? 

2d   place? 

Gth  place? 

10th  i)lai 


ce 


9th  place? 
8th  i)lace? 
2d  place? 
od    place  ? 


What  decimal  place  is  occupied  by  hundre<lths?  Tenths? 
Hundrcd-inillionths?  Thousandths?  Ten-thousandths?  Ten- 
nn'Ilionths?     Milliontlis?    r>illi()ntli>?    Hundred-thousandths? 


EXAMPLES  IN  NUMERATION. 

202.    1.  Read  the  decimal  l.iilG. 

ANALY8I8.--Tlie  figurm  of  the  dccimni  cxprem  2  tenths,  4  hun- 
dnnltliH  ami  6  thousandths,  which,  reducetl  to  cijnivalcnt  fractions 
having  a  coiuuiou  denominator,  Ix?comc  200  tliousnndllu*,  40  thou- 
sandths and  G  tiiousandths,  or  246  thousandths. 

The  whole  expression  is  read  4  and  24G  thousandths. 

Rule. — Read  Uie  decimal  as  an  intefjral  number  and  give  it 
ilie  deiwminaihn  of  Vie  right-liand  figure. 


Read  the  following: 

') 

.()84. 

14. 

.0231. 

26. 

4.16. 

o. 

.084. 

15. 

.4896. 

27. 

5.8406. 

4. 

.004. 

16. 

.3893. 

28. 

.60000. 

5. 

6.839. 

17. 

18.468. 

29. 

.00006. 

6. 

68.39. 

18. 

23.8009. 

30. 

.40508. 

7. 

683.9. 

19. 

649.3804. 

31. 

40.0004. 

8. 

.00450. 

20. 

.0020064. 

32. 

4000.004. 

9. 

3.02304. 

21. 

.4120465. 

33. 

518.6800. 

10. 

.050600. 

22. 

6.932474. 

34. 

4000.129. 

11. 

4.00008. . 

23. 

2.234006. 

35. 

80000.86. 

12. 

.000000856. 

24. 

3.000600. 

36. 

8000.086. 

13. 

1.000003894. 

25. 

4.006006. 

37. 

800.0086. 

NOTATION.  133 

•EXAMPLES  IN  NOTATION. 

203.    1.  Express  decimally  forty-thi-ee  thousandths. 

Analysis. — Since  43  thousandths  arc  equal  to  4  hundrcilthK  and  3 
thousandths,  we  write  3  in  thousandths'  place  and  4  in  hundnHhln' 
place,  and  as  there  arc  no  tenths,  0  in  tentlis'  place.  Hence,  forty-three 
thousandths  =  .043. 

Ruix. —  Write  the  numerator  of  tlie  decimal^  prefix  cipliers  if 
neceminj  to  indicate  the  deiwminatoi',  and  i)lace  the  decimal  point 
before  tenthn. 

Express  decimally; 

2.  Eight  teiitlLs.     Nine  tenths.     Five  tenths. 

3.  Two  liiindredtlis.    Eiglit  hundredths.    Six  hundredths. 

4.  Six  thousandths.     Four-hund red-two  thousandths. 

5.  Nine  ten-thousandths.    Eight  hundred  ten-thoiLsandtlis. 
-  6.  Seventeen  hundredths.     Fifteen  hinidred-thouKandths. 

7.  Forty-eight  tliousandths.     Five  hundred  tcn-millionths. 

8.  Ninety-three  ten-thousandths.     Ninety-three  billionths. 

9.  Filly-one  hundred-thousiuidths.    Fifty-one  thoiisiindths. 

10.  107  millionths.     306  ten-inillionths. 

11.  3251)  hundred-thousandths.      4268  h  und  red-mill  ion  t  lis. 

12.  420  ten-millionths.     3842  hundi-ed-thoubandths. 

13.  43(X)  billionths.     38496  hundred-billionths. 

14.  8.1  hundred-millionths.     8;")  hundred-billionths. 

15.  Six  thousiuid  ten-thousandths.     Five  ten-billionths. 
H).   Five  and  six-tenths.     Eight  and  nine  ten-thousandths. 
17.  Eighty  ten-thousiindths.     Forty  hundred-thousandths.. 

Express  decimally: 

18.  ^^.  22.  5t%V.  26.  64^. 

15>-  rhh'  23.  TTjWnr.  27.  SSjUU^ 

20.  r(f(AjW.  24.  ,1^.  28.  y^W- 

21.  ^tV  25.  4Ti4Ty.  29.  iVtfA. 


1.1  DECI  MAI.    IK  ACTIONS. 


In  rcatliiiL'^  expressions  of  United  States  currency,  the  cents, 
mills,  etc.,  mav  Im-  read  as  decimals  of  a  dollar.  Thus, 
$4.7:^:^5  ,nn    '  '        '  !   !  "       :-.^  cents,  or  84^Vi/W- 


KEDLCTION. 

CAsi;   1. 

204.  To  redncc  difiMiinilar  dc^rinialfi  to  Hiniilnr  dooi- 
iiialsi. 

1.  How  many  tenths  of  an  apple  are  there  in  1  apple  ?   How 
many  hundredths  in  10  apples?     How  many  thousandths? 

2.  How  many  hundredths  arc  there  in  6  tentli-^''     TTow 
many  thousandths?     How  many  ten-thousandths? 

3.  Express  6. hundredths  as   thousandths.     As   ten-thou- 
sandths.    As  hundred-thousandths.     As  milliontlis. 

4.  Express  8  thousandths  as  ten-thousandths.     As   luin- 
dred-thousandths.     As  millionths. 

205.  PlUNClPLE. — Annexituj   ciphers   to   a  decimal  dten   not 
alkr  its  value. 

WRIT  T  1    N      1    \  1    I:  (   /  s  /   > . 

1.  Reduce  .5,  .36,  .406  and  3.3109  to  similar  fractions. 

PROCESS.  .Vnalysis. — ^The  lowest  order  of  deci- 

__       5  000         mals  in  the  given  numbers  is  tcn-thoii- 
Bandthi^,  and  to  reduce  the  decimals  to 


.oO 


=    .3600 


similar  decimals,  we  must  change  them 


.406     =     .4060         all  to  tcn-tliousandths,  or  to  other  deci- 
3   3109  =  3   3109         mals  having  an  oriual  number  of  places. 

Since  annexing  ciphers  to  a  decimal 
does  not  alter  its  value,  we  give  to  each  number  four  decimal  places 
bv  annexing  ciphers,  and  this  renders  them  similar. 

Rule. — Give  to  all  the  given  iWiii^nU  fhp  mme   number  of 
decimal  places  by  annexing  ciphers. 


IIEDUCTION.  135 

2.  Reduce  .6,  .75,  .089,  to  similar  litrimals. 

3.  lieduce  .15,  .0406,  .0035,  .051,  to  similar  decimals. 

4.  Koduce  .0045,  .384t),  .51,  .51040,  to  similar  decimals. 

5.  lu'diico  3.35.  .345,  to  similar  decimals. 

Ivcdutc  ilio  lullowiiiu;  dipsiiuilar  decimals  to  similar  decimals: 


6.  .0436,  .04506,  .82. 

7.  .05,  4.825,  3.6046. 

8.  .3854,  .729,  8.053. 

9.  .8104,  .0008,  8000.4. 


10.  5,  .5,  .005,  50. 

11.  3.5,  .416,  .34,  14. 

12.  .214,  8.3,  .8,  4.6. 

13.  8.1,  43,  .68,  3.90. 


CASE  II. 
aOiy,  To  reduoo  a  decimal  to  a  coiiiiiioii  iVaetion. 

1.  If  5  tenths  be  written  as  a  common  fraction  what  will 
be  the  numerator?     What  will  be  the  denominator? 
*  2.  What  is  the  numerator  and  what  the  denominator  of 
the  decimal   18  hundredths,  when  expressed   as  a  common 
fraction  ? 

3.  Express  the  vahie  of  the  decimal  50  hundredths,  by  a 
common  fraction  in  its  smallest  terms. 

4.  Express  by  a  common  fraction  in  its  smallest  terms,  the 
following  decimals :  20  hundredths.  30  hundredths.  50  hun- 
dredths.    250  thousandths.     375  thousandths. 


U  niTTBN    EXEnCISES. 

1.  Reduce  .75  to  its  equivalent  common  fraction. 

PROCESS.  Analysis. —  .75  expressed    as   a   common 

75_.  7A^__|.         fraction  is  ^^  which,  being  reduced  to  it« 
.'smallest  terms,  equals  |. 

Rule. — Omll  the  chcimal  pointy  supply  ihe  denominator,  nvd 
reduce  the  fraction  to  its  loiveM  tenns. 


i.;< 


DECIMAL    INACTIONS. 


Retluce  the  following  decimals  to  equivalent  common  frac- 
tious in  their  smallest  terms: 


2.  .051. 

6.  4.0125. 

10.  .354. 

14.  .5675. 

3.  .03875. 

7.  .4355. 

11.  .00G25. 

15.  3.216. 

4.  .05625. 

M.  .0005. 

12.  .05375. 

16.  .4824. 

5.  .4375. 

"K). 

13.  .06501;. 

17.  .005396. 

18.  Reduce  .15|  to  an  equivalent  common  fraction. 

ruocKSB.  Analysis.— The  cxpres- 

.15,3^15|^1JA^^^  „       «on.l5fi8equal(ol_|},op 

100  100  ^'^  ^A        Reducing  tin.  .l.nom- 

100 

inator  aim  to  aevenths,  the  expression  becomes  f||,  or  f^^,. 
Change  the  following  to  equivalent  common  fractions,  or  to 


in 


ixed  numbers; 

19.  .12^. 

20.  .33i. 

21.  .161. 


22.  .87f 

23.  .04^. 

24.  .037^. 


25.  .562^. 

26.  .0031. 

27.  .078^. 


28.  .0003^. 

29.  2.756^. 

30.  13.81|. 


CASE  III 
207.  To  reduce  a  coniuion  rractioii  to  a  decimal. 

1.  One  half  of  an  apple  is  equal  to  li ow  many  tenths  of  an 
apple  ? 

2.  How  many  tenths  are  there  in    ^?    J?    |? 

3.  How  many  hundredths  are  there  in  J?    J?    }?    ^?    |? 

4.  How  many  hundredths  are  there  in  ^?    ^?    ^?    ^? 

5.  How  many  hundredths  are  there  in  ^,  or  100  hundredths 
divided  by  2?     How  many  in  :^? 

6.  How  many  luindre<lths  are  tlicre  in  4,  or  400  hundredths 
divided  by  5?     How  many  in  f  ? 

7.  How  many  thousandths  are  iIkiv  in  ^,  or  5000  thou- 
sandths divided  by  8?     How  many  in  |?     How  many  in  ^? 


IIEDUCTION. 


137 


WJITTT^N    EXEIICISES. 


1.  Reduce  j  to  an  equivalent  decimal. 

PROCESS.  Analysis. —  5 

8)5.000 

.625     Or, 

*  =  ««*  =  Tm  =  .625 


of  5. 


50 


tenths;  and  \  of  50  tenths  is  G 
tenths  and  2  tenths  remaining. 
2  tenths  are  eqnal  to  20  hun- 
dredths, and  \  of  20  hundredths 
is  2  hundredths  and  4  hundredths 
remaining.  4  hundredths  are  equal  to  40  thousandtlis,  and  \  of  40 
thousandths  is  5  thousandths.  Hence  f  is  equal  to  G  tenths  -}-  2  hun- 
dredths 4"  t>  thousandths,  or  .G25. 

Or  we  may  multiply  both  terms  of  the  fraction  by  1000  and  divide 
the  resulting  terms  by  8,  and  obtain  the  decimal  W(Ay>  9^  •C^''>' 

Rule. — Annex  ciphers  to  the  numerator  and  divide  by  Uie  de- 
nominator. Point  off  as  many  decimal  places  in  the  quotient  as 
tliere  are  ciphers  annexed. 

In  many  cases  the  division  is  not  exact.  In  such  instances  the  re- 
mainder may  be  expressed  as  a  common  fraction,  or  the  sign  -f  niay  be 
employed  after  the  decimal  to  show  that  the  result  is  not  complete; 
thus:  i---:.1662,  or  .166  4-. 

Reduce  the  following  to  equivalent  decimals : 


2.   i. 

8.   tV- 

14.    f 

20.    A. 

3.   i. 

9.   A. 

15.   -ft. 

21.^. 

4.   f 

10.  a- 

16.   ^. 

22.     f 

5.   |. 

11.  a. 

17.   ^,. 

23.    ^. 

6.   f 

12.   «. 

18.   H- 

24.    3V 

7.   }. 

13.   «. 

19.  tIt. 

25.    A. 

!hange  the  following  to  the 

decimal  form: 

26.  15f. 

29.  3.4^. 

32.  .87J^. 

35.  37.5^ 

27.  24|. 

30.  .23f. 

33.  .43}. 

36.  20.0f. 

28.  .82*. 

31.  .62^. 

34.  4.21f 

37.  .OOOJJ, 

138  l>iX;iMAL    1  iJACTlONS. 

ADDITION. 

20S.    1    What  i>  the  sum  of  ^^  and  t^?    -^  and  ^1    .3 

-.  \S  nui  IS  the  sum  of  -jV^r  and  itv ?  "Afe  ^^  VWy'  ^ - 
and  .20? 

3.  What  18  the  sum  of  j^  and  j^J  j^i^  and  y^§^? 
.005  and  .043? 

4.  Find  the  sum  of  ^  and  yj^.     Of  .5  and  .06.     .7  and 

T).   Fiii.l  the  sum  of  .6,  .31     '   '  t      Qf  .5,  .08  and  .006. 

209.  Principles. — The  pnnnjMes  are  the  mme  as  for  ad- 
dition of  integers. 

II  /;  /  /   /  f.  Y    EXBRCI8E8* 

1.  What  U  the  sum  of  .36,  2.136  and  4.5004? 

PROCESS.  Analysis.— We  write  the  numbers 

3g        ::^     .3600         ***  *^^*  units  of  the  same  order  shall 

2    IQA     2    1^60        ^^'^^tl  i"  the  Ranie  column,  and  add  as 

irciOA  4Pi0ftil         '"  integers,  separating  the  decimal  part 

_J_^ _J of  the  sum  from  the  integral  part  by 

6.9964       6.9964         **^^  decimal  point.     The  decimals  are 
made  similar  by  annexing  ciphers  until 
:ill  tlu'  till  imals  have  the  same  number  of  places. 

It  is  not  usual  to  make  the  decimals  similar,  for  if  they  are  written 
so  that  decimals  of  the  same  order  stand  in  the  same  column  it  is 
unnecessary  to  supply  tht-  lijilu  rs. 

llvi.i:.  —  TJi'-  nde  is  the  same  as  for  addition  of  integers. 

AVluit  i-  till'  sum  of 

2.  4.15,  3.86  and  .487?     I    5.  6.843,  48.25  and  17.286? 
.'>>.     n  0    -1  «4  and  .0507?  |    6.  .35,  .046  and  .00435? 
4  ^6  and  3.05?     j    7.   106,  .106,  1.06  and  10.6? 


>ri;i  iiAi HON.  139 

8.  Wliai  is  tlie  sum  of  $5.18,  $3.09,  8-10.  and  854.185? 

9.  Find  the  sum  of  $18.23,  $12.08,  $31,255  and  $6,625. 

10.  Add  $34.73,  $206,357,  $1200.18,  $3816  and  $137. 

11.  Express  as  decimals  and  add  6 J,  3f,  5 J,  6i  and  9 J. 

12.  A  laborer  earned  $7.25  in  one  week,  $7.12|  in  another, 
$9.18|  in  another,  and  $8f  in  another.  How  much  did  he 
earn  during  that  time? 

13.  What  is  the  sum  of  18  thousandths,  15  millionths,  81 
hundredths,  146  ten-thousandths,  834  hundred-thousandths? 

14.  What  is  the  sum  of  8  dollars  5  cents,  13  dollars  19 
cents,  18  dollars  3  cents  8  mills,  25  dollars  37  cents  5  mills, 
12|  dollars,  and  ^^g.  of  a  dollar? 

15.  Mr.  A.  paid  the  following  bills  for  repairs  upon  his 
premises,  viz:  carpenter-work,  $381.45;  plastering,  $215,385; 
plumbing,  $323.94;  and  other  expenses,  $181.57.  How  much 
did  he  pay  for  repairs  ? 

-16.  A  farmer  purchased  cloth  for  $13J,  boots  for  $8^^, 
crockery  for  $10|-i,  and  groceries  for  $15.49.  How  much 
did  he  pay  for  all  his  purchases  ? 


SUBTRACTION. 

210.  1.  From  f\  take  ^\.     From  .9  take  .5. 

2.  Find  the  difference  between  j^^  and  y^^;  -^j^  and  -j-J^; 
.19  and  .08. 

3.  Find  the  difference  between  jjpi^  and  xTnjT»   tijW  ""^ 

■nftpffJ  -^^^  ^^^  •^^^• 

4.  W^hat  is  the  difference  between  -^  and  yj^?     .5  and 
.06?     .7  and  .09? 

5.  What  is  the  difference  between  .16  and  .03?     .15  and 
.08?     .45  and  .3? 

211.  Principles. — The  pnncipl(i<  arc  Hit   .«iiiir  a.-<  j'>r  l/ie 
MiJUraction  of  integers. 


IIU  DECIMAL 


\-. 


irJlITTBir    EX  Bit  (   I  sis, 

1.  From  34.634  take  5.6857. 

PUOCE88,  Analysi*.      W       rite  tlic  numbew  8o  tli.it  iinit« 

34.0340  o^  tJ>e  same  order  sliall  stand  in  the  same  column, 

5.6857  and  subtract  as  in  intcgern,  separating  tlic  decimal 

.    28    9483  ^^^  ^^  ''**^  remainder  from  the  integral  part  by  a 

decimal  point. 

Off  In  the  first-  procem  the  decimaln  are  made  simi- 

o  J    /»  rt  4  lar  by  annexing  a  cipher  to  the  minuend. 

5*6  8*1  7  In  the  second  pnxress,  which   i«  the  otu    r-ni- 

1 monly  employed,  the  cipher  i<  ixt  written,  but  w© 

28.9483  Ruppoee  it  to  be  annexed. 

KuLE. — The  ride  is  (he  same  as  for  subtraction  of  integers. 

(2.)  (3.)  (4.)  (5.) 

From        48.356        39.82  343.25  8118.375 

Subtract  23.453         13.856        818.375        8  43.50 

6.  What  is  the  difference  between  .7134  and  .50645? 

7.  What  is  the  difference  between    8.34  and  6.3168? 

8.  What  is  the  difference  between     100  and  .03846? 

9.  From  84  millionths  take  84  ten-mill  ion  ths. 
10.   From  80  thousand  take  80  thousandths. 

11.-  From  29  dollars  3  cents  take  17  dollars  9  cents. 

12.  From  27  dollars  8  cents  take  9  dollars  37  cents  5  mills. 

13.  If  I  spend  845. 89 J  for  merchandise,  how  much  will  I 
have  left  after  jiaying  for  it  with  a  fifty-<lollar  bill  ? 

14.  A  gentleman's  income  was  812384.16,  and  his  expenses 
the  same  year  were  89864.18.  How  much  of  his  income  was 
left? 

15.  The  receipts  of  a  reaper  manufactory  for  the  year  1876 
Avere  81374837.64,  and  the  expenditures  81298369.58.  What 
was  the  surplus? 


MULTI  l»LICATION.  1 4 1 


Ml   I/ri  PLICATION. 

2X2.    1.  Wliatisthei)r()(liictof/(^X2?    -j^XS?    .4x2? 

2.  How  many  decimal  figures  arc  there  in  the  product  of 
tenths  by  units? 

3.  Wiuit  is  the  product  of  -j J^  X  4 ?    t?(T  X  4 ?    .04  X  2  ? 

4.  How  many  decimal  figures  are  there  hi  the  product  of 
hundredths  by  units? 

5.  What  is  the  product  of  j\X-y\^    AXtV?     .4  X  .2? 
().  How  many  decimal  places  are  required  to  express  the 

product  of  tenths  multiplied  by  tenths? 

7.  What  is  the  product  of  V^X-rfir^    tVXtJtf?    .4X.02? 

8.  How  many  decimal  places  are  required  to  express  the 
l)roduct  of  tenths  multiplied  by  hundredths  ?  Tenths  by  thou- 
ijuudths?     Hundredths  by  thousandths? 

0.  If  the  multiplicand  has  two  decimal  places,  and  the  mul- 
tiplier three,  how  many  will  there  be  in  the  product? 

10.  How  does  the  number  of  places  required  to  express  the 
pnHluct  of  two  decimals  compare  with  the  number  of  decimal 
l)laccs  in  the  factors? 

213.  PrincipI-E. — The  product  of  ttvo  decimals  contains  as 
many  decimal  places  as  Uiere  are  decimal  places  in  both  factors. 

WRITTEN    EXERCISES, 

1.  Multiply  .312  by  .24. 

PROCESS.  Analysis.— .312  X  -24  =  iVi^  X-AS  =  jlUh  = 

.312  •^"^^^-    ^^^^^  '^^  2  X  .24  =  .07488.    Or, 

cy  t  We  may  multiply  as  if  the  numlKTs  were  integers; 

— '- and  since  the  multiplicand  has  three  decimal  places, 

1-^40  and  the  multiplier  two  places,  the  product  must  have 

62  4  five  places.     (Prin.)     Or,  thousandths  multiplied  hy 

.07  488  hundredths  give  hundred-thousandths,  the  denomi- 
nation  of  tlic  product. 


142 


lilLK. Multiply  i,.^    ,j   ill'    immu,,.^   Mr/r    ,,ii'<jfi.^,   utiiiJi'Om  Hie 

right  of  the  prcKlnd  jmid  offoA  many  figures  for  dcciviaU  as  there 
are  decimal  places  in  both  factors. 

If  the  product  does  not  contain  as  many  figure8  as  thorc  are  decinjalH 
in  botl>  factors  the  deficiency  must  be  supplied  by  prefixing  ciphers. 


Mult  Mil  V 


8. 
4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 


.45 
.436 

348 
3.48 
34.8 
8.75 
.579 

486 
2.48 
3.94 


by  .34. 
by  4.: 
by  .4tl. 
by  .44. 
by  .64. 
by  .74. 
by  8.5. 
by  .035. 
by  3.75. 
by  2.37. 
by  3.84. 
hy  .0064. 


Multiply : 


14. 

m. 

17. 

IS. 

19. 
20. 
21. 
22. 
23. 
24. 
25. 


S2.75 

S31.16 
34.165 

3.845 
8ir,.i8 
500.15 
37.856 

70.05 

63.18 
64.032 

51.27 
J  of  .55 


by5t. 
by3f. 
by  7.3. 
by  .666J. 
by  5.36. 
by  30.04. 
by  .0405. 
by  2.308. 
by  .0634. 
by  5.321. 


hv 


.f  0.5. 


26.  Multiply  4.639  by  100. 


PROCESS. 

4.639 

100 
463.900 


Analysis. — Since  each  removal  of  a  figure  one 
place  to  the  left  increases  its  value  tenfold,  the 
removal  of  the  decimal  point  one  place  to  the  right 
multiplies  by  10,  and  the  removal  of  the  point  two 
places  to  the  right  multiplies  by  100.     Hence, 


Rule. — To  vndfiply  by  1  with  any  number  of  ciphers  annexed^ 
remove  the  decimal  point  as  many  places  to  Hie  right  as  there  are 
cipJicrs  annexed. 

27.  Multiply  384.64  by  100.     By  10.     By  1000. 

28.  Multiply  1.8465  by  100.     By  1000.     By  10000. 

29.  AVhat  will  be  the  cost  of  34.5  yards  of  cloth  at  $3.15 
])er  yard  ? 


DIVISION.  143 

30.  When  land  is  worth  3137.18  \)er  acre,  how  much  must 
be  jjaid  for  a  farm  of  38  acres? 

31.  Since  16.5  feet  make  a  rod,  how  many  feet  are  there 
in  23.7  rods? 

32.  What  will  be  the  cost  of  9  houses  at  $3847.93  each? 

33.  When  wheat  is  worth  ?1.62|  per  bushel,  what  will  37.3 
bushels  cost? 

34.  What  is  the  value  of  57  barrels  of  flour  at  $8.37^  a 
barrel? 

35.  Mr.  Orr  sold  8  horses  at  an  average  price  of  $213.27 
each.     How  much  did  he  receive  for  them  ? 

36.  A  lady  made  the  following  purchases,  viz:  37  yards  of 
bleached  sheeting  at  8  AS^  per  yard,  8  yards  of  velvet  ribbon 
at  S  .37^  per  yard,  27  yards  of  silk  at  32.35  per  yard.  What 
was  the  entire  cost  of  her  purchases? 


DIVISION. 

214.    1.  What  is  the  product  of  .6  by  8? 

2.  4.8  is  the  product  of  two  factors,  one  of  which  is  8: 
what  is  the  other  factor? 

3.  What  is  the  product  of  .0  by  .8? 

4.  .48  is  the  product  of  two  factors,  one  of  which  is  .6: 
what  is  the  other  factor? 

5.  What  is  the  product  of  .06  by  .8? 

6.  .048  is  the  product  of  two  factors,  one  of  which  is  .06* 
wliat  is  the  other  factor? 

7.  What  is  the  product  of  .06  by  .08? 

8.  .0048  is  the  protluct  of  two  factors,  one  of  which  is  .06: 
what  is  the  other  factor? 

9.  How  many  decimal  places  are  there  in  the  quotient  when 
tenths  are  divided  by  units?  Hundredths  by  tenths?  Thou- 
sandtlis  ])V  Inindndtlis?      T«-n-lli(>u>:m(lt]is  l.v  lumdrt'dths? 


1  1  I  i>i:<  I  M  \  I     1  I.  \<  1  1-  ..N-. 

10.  How  iii.iiiy  <icciiiml  places  are  there  in  the  product  of 
any  two  lli.  t,,i^ ; 

11  If  the  product  and  one  of  two  fectors  are  given,  how 
may  tiie  iiund)er  of  decimals  in  the  other  factor  be  found? 

12.  How  may  the  factor  be  found? 

13.  Since  the  factor  sought  will  be  the  quotient,  bow  many 
decimal  places  will  there  be  in  tlu-  quotient? 

215.  PRINXIPLE. — T/ie  quotient  wiU  c<mtain  as  many  deci- 
vHil  j)lae€$  as  Uie  number  of  decimal  plaoet  in  the  dividend  ex- 
ceeds those  in  Ute  divisor. 


WJllTTr  Y    r  T  7-  /.  .    /  >  /.  > . 

1.  Divide  8.88  by  2.  1 

PROCESS.  An  A  LYSIS. —8.88  -i-  2.4  =  fJJ  -4-  ^J  =  m 

2.4)8.88(3.7  Xi5  =  H  =  3.7.    Or, 

y  2  ^®  divide  aa  if  the  nMmlx>rs  were  inte- 

_  Re»*;  «nd  pince  the  dividenii  has  two  deci- 

•*  "  "  mal  place!),  and  the  divisor  one,  the  quotient 

168  winimv..  ..n..     I  p.;  Ml 

Rule. — Divide  as  if  he  nninnfis  u,,,-  tiiinjrr!<,  ami  from  the 
rigid  of  Uie  quotient  point  off  as  many  figures  for  decimals  as  Hie 
number  of  decimal  places  in  the  dividend  exceeds  the  number  of 
those  in  Hie  divisor. 

1.  If  the  quotient  does  not  contain  a  Rufficient  number  of  decimal 
places  the  deficiency  must  be  supplied  by  prefixing  ciphers. 

2.  Before  commencing  the  division,  the  number  of  decimal  placis 
in  the  dividend  sliould  be  made  at  least  equal  to  the  number  of  deci- 
mal places  in  the  divisor. 

8.  When  there  is  a  remainder  after  using  all  the  figures  of  the  divi- 
dend, annex  decimal  ciphers  and  continue  the  division. 

4.  For  the  ordinary  purposes  of  business  it  is  not  necessary  to  carry 
the  division  further  than  to  obtain  four  or  five  decimal  figures  in  the 
quotient. 


DIVISION. 


1  i". 


♦  (TV  J 


Divide : 

:l.      2.450  by  9.8. 

:.  .00335  by  .67. 

1  I ;  2512  by  3.7. 

5.  .05475  by  15. 

0.  18.312  by  .24. 

7.  105.70  by  3.5. 

H.  .11928  by  .056. 


16.  Divide  423.68  by  100. 


Divide : 

9.     .04905  by  .327. 

10.  135.05  by  .037. 

11.  687.50  by  .025. 

12.  34.368  by  .013. 

13.  .014532  by  .0692. 

14.  71.142  by  .0071. 

15.  .027538  by  .0326. 


PROCESS.  Analysis. — Since  each  removal  of  a  figure 

1  00)423.68         o"^  place  to  the  right  decreases  its  value  ten- 

1   9  Q  ft  A         ioXa,  the  removal  of  the  decimal  jwint  one  place 

to  the  left  divides  by  10,  and  the  removal  of 

the  decimal  point  two  places  to  the  left  divides  by  100.    Hence, 

■Rule. — To  divide  by  1  with  any  mimher  of  cijyJiers  annexed^ 
remove  Hie  decimal  point  as  many  lylaces  to  (he  left  as  Hiere  are 
ciphers  annexed. 


Divide : 

17.  48.26  by  100. 

18.  382.457  by  1000. 

19.  13.8542  by  1000. 


Divide: 

20.  4.897  by  100. 

21.  .06045  by  1000. 

22.  3845.63  by  10000. 


23.  At  $8.25  per  ton,  how  much  hay  can  be  bought  for 
$29.35? 

24.  At  $.18  per  dozen,  how  many  dozen  eggs  can   be 
Iwugbt  for  $32.40? 

25.  If  I  pay  $106.40  for  nr>  hat-,  hnw   uuu-h  do  lluv  rarh 
cost  nie? 

26.  How   many  hog.«<ht'ads  of  mohi-s<  <.   at    >"»7.38  each, 
can  be  purclm.«ed  for  $1319.74? 

,27.  How  many  stoves,  at  $21.35  each,  can  be  bought  for 

10 


146  Di:<  i\i  \ '    '  '■  ^'  I  '"Ns. 


SHORT  PllOCESSES. 

21(>.  Many  methods  have  been  devised  for  abhreviating 
tlu  procojis^s  of  computation,  among  which  the  following  are 
of  much  practical  value: 

CASE  I. 

217.  To  iniiltiply  by  a  iininbor  a  little  less  than  a 
nil  it  of  the  next  higher  order. 

1.  li'.w  much  less  than  10  is  9?  Than  100  is  99?  Than 
1000  is  999?  Than  100  is  98?  Tlum  100  is  97?  Than 
100  is  96? 

2.  How  much  less  than  10  times  a  number  is  9  times  the 
number?  liaii  100  times  is  99  times?  Than  1000  times  is 
999  times?  Than  100  times  is  97  times?  Than  1000  times 
is  997  times? 

ir  n  T  T  T  r  \    rrrnrrsrs. 
1.  Multiply  4685  by  97. 


Analysis. — Ninety -seven 
468500=  100  times  4685       times  a  number  is  1(X)  times 
14055=       3  times  4685 


the  number  minus  3  times 
the  number.     We  annex  two 


454445=      0  7   liiu' -    H*.  S5       ciphers  to  the  muhiplicand, 

thus  muhiplying  by  100,  and 
tlun  subtract  from  the  product  three  times  the  multiplicand,  leaving 
97  tiuKs  the  nmltiplicand. 


Multiply : 

2.  3856  by  99. 

3.  4832  by  998. 


Multiply: 

6.  34G725  by  997. 

7.  486965  by  97. 


4.  48567  by  !•'  8.  843256  by  96. 

5.  89736  bv  1)^.  .  9.  586436  by  9996. 


SHORT   PROCESSES. 


147 


10.  AVhat  will  be  the  cost  of  385  bushels  of  corn  at  3  .97 
per  bushel? 

11.  How  much  must  be  paid  for  1373  pounds  of  tea  at 
S  .96  per  pound? 

12.  At  897  Y>er  acre,  what  will  a  farm  of  139  acres  cost? 

CASE  II. 

218.  To  multiply  iiheii  one  part  or  the  multiplier  is 
a  flietor  oi*  another  part. 

1.  Multiply  4  by  8;  4  by  8  tens;  4  by  16  tens. 

2.  Multiply  7  by  6;  7  by  6  tens;  7  by  18  tens. 

3.  AVhen  7  times  a  number  is  known,  how  may  21  times 
a  number  be  found?    21  tens  times?    21  hundreds  times? 


WRITTEN    EXEBCiaJSa, 


1.  Multiply  3684  by  124. 


PROCESS. 

3684 

124 

14736 

44208 

456816 


Analysis. — The  multiplier  may  be  regarded  aa 
composed  of  12  tens  and  4  units,  or  3  times  as 
many  tons  as  units.  AVe  therefore  first  muUiply 
by  4  units;  and  since  there  are  3  times  as  many 
tens  as  nnit-*,  we  multiply  this  product  by  3,  and 
write  the  result  as  tens  by  placing  it  one  place  to 
the  left  of  units. 


Multiply : 

2.  3825  by  63. 

3.  5973  by  93. 

4.  8126  by  123. 

5.  6924  by  213. 

6.  14273  by  246. 

7.  28653  by  328. 

8.  68435  bv  217. 


Multiply : 

9.  3692  by  357. 

10.  6384  by  248. 

11.  4239  by  369. 

12.  12783  bv  189. 

13.  36412  by  279. 

14.  36485  by  2408. 

15.  2975?^  bv  3609. 


148 


J>K<;iMA 


\<  lluNS. 


Is2J? 
Is  25? 


Is  50? 


CASE  III. 

219.  To  malflply  by  a  uniiilM^r  nliicli  In  an  aliqaot 
part  of  Moiii€*  liiglier  unit. 

1.  Wimt  jMirt  of    10  is    2?     U    o'i 

2.  What  part  of  100  is  10?    Is  20? 
12i?     Is  831? 

3.  What  ixirt  of  a  dolhir  are  50  cents?     25  rmt?'     20 
cents?     10  conts?     12V  rontK? 

4.  Wha  33J  cents?     l^  cents? 

220.  An  Aliquot  l*avt  of  a  number  is  such  a  part  as 
will  exactly  divide  the  number. 

Thus,  5,  10,  12J,  etc,  arc  aliquot  parts  of  100. 

The  aliquot  jwrts  of  10,  commonly  used,  are: 
6  =  iofl0.        I      3J:=ioflO.      I    2|=JoflO. 

The  aliquot  parts  of  100,  commonly  used,  are: 


50  =  i  of  100. 
25  =  :J  of  100. 
20  =  i  of  100. 


33i=:Jof  100. 
16}  =  ^  of  100. 
12i  =  ^of  100. 


Otlier  parts  of  100  are; 


40  =.  I  of  100. 
60  =  ^  of  100. 
80  ^-  i  of  100. 


37^  =  1  of  100. 
64-=  I  of  100. 
87^  =  J  of  100. 


10  =  TVof  1^- 

8^  =  ^  of  100. 
6J  =  3J^of  100. 


66}  =  I  of  100. 

75=  I  of  100. 

43}  =  tt  of  100. 


WniTTBN     EXERCISES, 


1.  Multiply  434  by  25. 

rROCESS.  Analysis.— Since  25  is  \  of  100,  we  may  mnl- 

4)43400         tiply  by  25  by  first  multiplying  by  100,  and  then 
108  50        taking  \  of  tlie  prmhict. 


MIUKT    PROCESSES.  149 

2.  What  will  ^5  yards  of  cloth  cost  at  3  .33 J  per  yard? 

PROCESS.  Analysis.— At  §1  per  yard,  the  cloth  would 

3)85  cost  $85;  and  at  $.33 J,  or  J  of  a  dollar,  it  will 

"28    334-        cost  i  of  $85,  or  $28.33 J. 


Multiply: 

3.  688  by  12f 

4.  402  by  16J. 

5.  5056  by  25. 

6.  75630  by  33^. 


Multiply : 

7.  8404  by  50. 

8.  2160  by  37^. 

9.  4236  by  66|. 
10.  7288  by  75. 


11.  What  will  be  the  cost  of  27  yards  of  cloth  at  $  .25  per 
yard? 

12.  When  butter  is  worth  S^  cents  a  pound,  what  will 
824  ix)unds  be  worth? 

13.  What  will  be  the  cost  of  216  pounds  of  tea  at  $  .75  per 
pound  ? 

14.  AVliat  will  287  bushels  of  oats  cost  at  37^  cents  per 
bushel  ? 

15.  What  will  394  bushels  of  potatoes  cost  at  62^  cents 
per  bushel? 

16.  What  is  the  value  of  319  bushels  of  wheat  at  $1.37J 
per  bushel? 

CASE  IV. 

22\,  To  fliKl    the   cost  ^lioii    the   quantify  and   the 
price  of  KM)  or  1<MH>  are  Riven. 

1.  When  the  cost  of  100  articles  is  known,  how  can  the 
cost  of  500  l)e  found?     600?     800?    900? 

2.  When  the  cost  of  100  articles  is  given,  how  can  the 
cost  of  250  l)c  found?    350?    750?    850?    950? 

3.  How  many  times    100  is    250?      275?       280?      285? 

4.  How  many  times  1000  is  3000?     3500?     3750?     4585? 

4t 


160  DECIMAL   FRACTIONS. 


WRITTEN    BX  I    i:  <    isi:s, 

1.  What  will  be  the  cost  of  375  pounds  of  fish  at  16.76 
per  100  pounds? 

PROCESS.  AvALYai8.--8ince  100  pounds  co«t  $0.76,  375 

$6.76        pouuda,  or  3.75  times  100  ix>und«,  will  cost  3.75 

3   yg         tim«i  $6.76,  or  $25.31^.    Or,  the  price  may  be 

multiplied  by  the  quantity,  nnd  thedeeiuiul  point 


$26.3126        removed  two  places  to  the  left  in  the  product. 

The  letteni  C  and  M  are  used  instead  of  the  wordR  hundiri   mk! 
(AoMKtfu/,  rc»pcrtively. 

2.  How  i!;!;(  !i  will  6075  pounds  of  coal  cost  at  $.35  per 

liuixlrt  (l-wi 

:\.   When  suingies  cost  $4.75  per  M,  how  much  will  8^00 
shiiii:!"  -  co.-t? 

1.    Wi.;i;    i^   il:-   pi-ir,.   ,.r  a  l-.a.]  .,f  Lay  wci^L^liing  1925 
pmiihl-,  at  _?^t>..jO  ptT  !  ■!!     '_'<MM»  poundsj^y 

>.   What  is  the  o'^i  ,<\'  1»;795  pounds  of  plaster  at  $4.50 
per  100  pounds? 

6.  How  much  will  12l*7*i5  laths  cost  at  82.75  per  M? 

7.  What  is  the  wst  of  6975  bricks  (a;  $3.25  j^er  M? 

8.  What  i-  ilu  cost  of  1825  pounds  of  iron  @  $45  i)er  ti^m? 

9.  Tlnw   much  iiui>t  be  jmid  for  6780  envelope?  (^1  82.75 
per  yV: 

in.   What  would  be  the  cost  of  550  pine-apples  at  $13.25 
per  C? 

11.  ^\Tiat  will  bo  the  cost  of  1592  pounds  of  beef  at  $4.50 
per  hundivd  jx)unds? 

12.  What  will  15000  pounds  of  coal  cost  at  87.50  a  ton? 
!:>.  What  will  be  the  cost  of  2294  pounds  of  broom-corn 

at  $55  jx^r  ton? 

14.  What  will  be  the  cost  of  1964  pounds  of  maple-sugar 
at  $13.45  per  hundred-weight? 


ACCOUNTS   AND   BILI^. 


151 


ACCOUNTS  AND  BILLS. 

222.  A  Debt  is  an  araoimt  which  one  person  owes  to 
another. 

223.  A  Credit  is  an  amount  which  is  due  to  a  person, 
or  a  sum  paid  towards  discharging  a  debt 

224.  A  Debtor  is  a  party  owing  a  debt. 

225.  A  Creditor  is  a  party  to  whom  a  debt  is  due. 

226.  An  Account  is  a  record  of  debts  and  credits  be- 
tween parties  doing  business  with  each  other. 

227.  The  Balance  of  an  Account  is  the  difference 
between  the  amount  of  the  debts  and  credits. 

228.  A  Bill  is  a  written  statement  given  by  the  seller  to 
the  buyer,  of  the  quantity  and  price  of  each  article  sold,  and 
the  amount  of  the  whole. 

220.  The  Footing  of  a  Bill  is  the  total  cost  of  all 
the  articles. 

230.  A  bill  is  Iteceijited  when  the  words  Received  Pay- 
ment are  written  at  the  bottom,  and  the  creditor's  name  is 
signed  either  by  himself  or  some  authorized  person. 

231.  The  following  abbreviations  are  in  common  use: 


®, 

At. 

Do., 

The  same. 

Mdse., 

Merchandise. 

%, 

Account 

Doz., 

Dozen. 

No., 

Number. 

Acc't, 

Account. 

Dr., 

Debtor. 

Pay't, 

Payment. 

Bal., 

Balance. 

FVt, 

Freight 

Pd., 

'  Paid. 

Bl)l, 

Barrel. 

Hhd., 

Hogshead. 

Per, 

By. 

Bo't, 

Bought 

Inst, 

This  month. 

Rec'd, 

Received. 

Co., 

(  ompany. 

Int, 

IntercHt. 

Yd.. 

Yard. 

Cr, 

Creditor. 

Lb., 

Pound. 

Yr., 

Yemr. 

152 


DECIMAL   FIl ACTIONS. 


<^" 


t 


I  I         I  I 


^  ^  V  *^ 


> 


^^  ^ 


^^ 


^:^ 


v>i    NS    f^^ 


« 


^S^ 


<^ 


'>t>  \-> 


X 


ACCOLM.S    A.ND    JilLI,S.  153 

Copy,  iill  (Mit  and  find  tlic  footings  of  ouch  of  tho  follo\viu<r: 
(2.) 

RocHJ^STER,  March  1,  1877. 
Mi;.  .1.    11.   A  HAMS, 

Bought  of  Howe  &  Rogers: 


75J  yards  of  Carpeting    . 

37  yards  of  Drugget  .     . 

8  Rugs 

5  Mata 

18  yards  Oil-(loil».     .     . 

9  yards  Carpet  Lining 
3  Carpet-sweepers  .  . 
2  doz.  Stair-rods .     .     . 


(a)    $2.12A 
1.20 
4.1G 

-       2.:i7i 

1.08 
.12.^ 
■J.OO 
-.25 


Received  Payment, 

Howe  &  Rogers. 


(3.) 


Memphis,  May  20,  1877. 


Mr.  George  B.  Sherman, 

To  Samuel  B.  Smallwood,  Dr. 


To  37  bbl.  Pork 

"  127  bbl.  Flour 

"      3  hhd.  Molasses— 169  gal 

"  29  firkins  Butter— 2120  lb 
"      3  boxes  Raisins    .     .     . 

"  5  bbl.  Kerosene — 207  gal 

"  25  doz.  cans  Fruit  .     .     . 
"      3  packages  T..1.M.'.-..—:^.l ft  n 

"  13doz.Spio.^ 


(5)  $24.35 

"       8.15 

.43 

.31 

•       .isi 
•j.io 

.15 
1.10 


lleceived  Ihtpnenl  hy  note  at  60  rfay«, 

Sam'l  B.  Sm.\llwood. 


154 


DECIMAL.   FRACTIONS. 

(4.) 


Mh.  Erastus  p.  Ga  rr/i. 


New  York,  AprU  1,  1877. 
To  Sturdevant  &  Co.,  Ih-. 


1»77. 

II 

Jan. 

0 

u 

13 

Feb. 

3 

u 

15 

Mar. 

8 

(1 

12 

1877. 

Jan. 

24 

Feb. 

10 

.    n 

18 

To  3  Gold  Watches— $124.50,  $61.24,  $57.18 

a   $1.15 

13.10 
51. 

"     12. 

"      12.50 


"  437  pwt.  Gold  Chains 

**  35  seta  Plated  Tea-«er\ 

tt  <T      i(                  II                          II 

"  5  Silver  Pic-kniTes     . 

"  12  Plated  Ice-pitchers  . 


Cr. 

By  Cash 

"    Draft 

"    Mdse.  returned  .     . 


$21 
327 

78 


How  much  is  still  due  Sturdevant  &  Co.? 

Make  out  in  proper  form  and  receipt  the  following: 

5.  Mrs.  M.  T.  Dana  bought  of  G.  C.  Smith  &  Co.,  25  yd. 
of  calico  (n)  10  cents,  37  yd.  of  sheeting  @  18J  cents,  2  pairs 
of  gloves  @  $1.50,  1  sun-umbrella  @  $6.75,  5  yd.  of  Ham- 
burg edging  @  25  cents,  7  pairs  of  hose  @  $  .85. 

6.  ISIr.  C.  C.  Lovell  bought  of  R.  P.  Lawton  7568  feet  of 
hemlock  0  $12.75  per  M,  8539  feet  of  pine  flooring  @  $23.50 
per  M,  5608  feet  of  clear  pine  @  $45  per  M,  3815  feet  of  oak 
joists  @  $32  per  M,  7346  feet  of  ash  flooring  @  $34  per  M. 

7.  Mr.  George  M.  Line  bought  of  Steele  &  Avery  15  reams 
of  commercial  note  paper  @  $1.25,  7500  envelopes  @  $3.65 
per  M,  18  gross  steel  ix^ns  @  $  .75  per  gross,  24  Ridi^ath's 
Histories  @  $1.25,  9  Webster's  Dictionaries  ©  $10.25. 


1>E<IMAL    FRACTIONS.  155 


REVIEW   EXERCISES. 

1.  A  farmer  sold  his  butter  at  34  cents  a  pound,  and 
received  for  it  $123.59.     How  many  jwunds  did  he  sell? 

2.  A  gallon  of  distilled  water  weighs  8.339  pounds.  How 
much  will  15 J  gallons  weigh? 

3.  A  square  rod  contains  272J  square  feet.  How  many 
square  feet  are  there  in  7f  square  rods? 

4.  The  best  anthracite  coal  is  said  to  weigh  55.32  pounds 
per  cubic  foot.  How  many  cubic  feet  will  weigh  a  ton  of 
2000  pounds? 

5.  The  number  of  cubic  inches  in  a  bushel  is  2150.42. 
How  many  cubic  inches  are  there  in  1000  bushels? 

6.  What  is  the  quotient  when  3  is  divided  by  3  thou- 
^ndths? 

7.  What  is  the  quotient  when  300  is  divided  by  3000 
hundred-millionths  ? 

8.  A  lumber  merchant  had  2182565  ft.  of  lumber.  After 
selling  .20,  or  20  per  cent.,  of  it,  he  lost  15  per  cent,  of  the 
remainder  by  fire.    How  many  feet  of  lumber  were  burned? 

9.  What  will  385  pounds  of  flour  cost  at  $4.25  per  hun- 
dred-weight? 

10.  At  3  .11:J^  per  pound,  how  many  pounds  of  sugar  can 
be  bought  for  831.25? 

11.  Bought  26  yards  of  broadcloth  at  S4.37|  per  yard, 
and  [Miid  for  it  in  pork  at  $7.25  per  hundred- weight.  How 
much  pork  will  it  take  to  pay  for  the  cloth? 

12.  If  15  tons  of  hay  cost  $125.25,  what  will  35  tons  cost? 

13.  If  Rid  path's  histories  retail  at  $1.25  each,  what  will 
be  received  for  350  sold  at  that  rate? 

14.  When  pork  is  selling  at  $6.25  per  hundred-weight, 
how  much  can  he  l)ought  for  $325? 

15.  When  8000  is  divided  by  .004,  what  is  the  quotient? 


156  DECIMAL    FRACTIONS. 

K;.    \\\u\\  .0008  is  (lividr,!  I.y  jniMMi,  \J,;,t  i..  I  he  quotient? 

17.  Hmu  many  .la\~  nniM  a  la  i-k  at  $1.37|  per 
day,  t(»  i»ay  l<)r  ^  curds  of  wood  at  -"ri.  i-J;^  per  cord? 

18.  A  lady  bought  the  following  articles:  27  yards  of  silk 
at  S2.75  iKjr  yard,  11  yards  of  lace  at  86. 37 J  per  yard,  9 
pairs  of  gloves  at  82.15  per  pair,  10  jmirs  of  hose  at  81.10 
jx'r  pair.     What  was  the  amount  of  tlie  ])urchase? 

!'•.  Ill  man  earns  812^  per  week,  and  spends  87|  per 
week,  in  how  many  weeks  can  he  save  8500? 

20.  What  is  the  value  of  iioloO  bricks  at  87.25  per  M? 

lM.  ^\  Iiai  i^  ih.  value  of  a  farm  of  195  acres  if  91  acres 
ar«  WMiih  x.hs.s.Oo,  and  the  remainder  81.12A  per  acre  more? 

L'_*.  A  (hover  l)ought  375  sheep  at  84.50  per  head.  He 
sold  200  of  til.  Ill  at  a  loss  of  $.20  per  head,  and  Laiia.l 
enough  on  the  re.^L  to  l^nlanrc  the  ln>>-.  What  du\  la  -(  t 
l)er  head  for  the  rest? 

23.  The  exjjenses  of  conducting  a  business  enterprise  were 
.40  of  tlie  entire  profit?.  If  the  profits  were  .15  of  the  value 
of  the  LI. X Ills  sold,  how  luiuh  was  received  from  the  sale  of 
•ro.'.ls  it  ih(    pi.. lit-  were  89000  more  than  the  e 

_M.    l.\p!-.'ss  as  a  decimal    .^- — —  ^- -• 

25.  A  s])f('ulat..r  l)oiiLi-ht  ."XldO  luishcl-  >  l'.'  in  al  >  .I'l.".  jur 
Imsliel.  He  sold  .25  of  it  i". .r  >^  .70  j)er  bushel,  and  the  re- 
main.hr  1. n-  such  price  that  la-  italized  a  ])rofit  on  the  whole 
ot'  >  147. .")<>.  lldw  much  (lid  lie  Lie  t  \n  V  hushcl  for  the  re- 
mainder? 

lM>.  'Vhc  estimated  value  of  ^Ir.  A.'s  firm  \va<  -^tioOO.  If  he 
s.d.l  a  p.ii'ti.Ti  .•!'  it,  at  it<  estimated  value  per  acre,  for  82275, 
what  decimal  part  of  the  farm  did  he  sell? 

27.  A,  B  and  C  divide  645i  bushels  of  wheat  among  them- 
selves. A  tak.-  .•■'.7',,  J>  ■]..  and  C  the  remainder.  How 
manv  l)uslul.-  had  each? 


DEFINITIONS. 

232.  A  Concrete  Ntunber  is  a  number  used  in  con- 
iie<nion  with  some  specified  tiling. 

Thus,  5  books,  7  trcca,  8  horses,  are  concrete  numbers. 

233.  An  Abstract  Nuniher  is  a  number  that  is  not 
used  in  connection  with  any  specified  thing. 

Thus,  5,  7,  8,  are  abstract  numbers. 

234-.  A  Denonihidte  Ninnher  is  a  concrete  number 
in  which  the  unit  of  measure  is  estr-blished  by  law  or  custom. 
Thus,  5  yards,  3  feet,  7  pounds,  3  ounces,  are  denominate  numbers. 

235.  A  Simjtle  Denominate  Number  is  a  denom- 
inate number  composed  of  units  of  the  same  denominations. 

Thus,  5  feet,  9  pounds,  3  miles,  are  simple  denominate  numbers. 

236.  A  Compound  Denominate  Number  is  a 

denominate  number  composed  of  units  of  two  or  more  denom- 
inations which  are  related  to  each  other. 

Thus,  6  feci  and  4  inches,  8  hours  and  32  minutes,  arc  compound 
denominate  numl)cr8. 

237.  A  Standard    I'nit    is  a    unit  of  measure   from 
which  the  other  units  of  the  same  kind  may  ha  derived. 

Thus,  the  yard  is<host:m(l:inl  unit  fnun  wliioh  all  measures  of  length 
are  fornu'd;   tlu>  Trov  poiiiul  tlu-  stand.'ird  unit  of  wilirlit. 


158  DENOMINATE    NUMBERS. 

238.  A  Scale  is  the  ratio  by  which  numbors  increase  or 
decrease.     Scales  are  either  uniform  or  varyintj. 


MEASURES  OF  VALUE. 

239.  Money  is  the  measure  of  value. 

It  is  also  called  Oarreneyt  and  is  of  two  kinds,  viz:  coin 
and  paper  money. 

240.  Coin  or  Specie  is  stamix?d  pieces  of  metal  having 
a  value  fixed  by  law. 

241.  Paper  Money  is  notes  and  bills  issued  by  the 
Government  and  banks,  and  authorized  to  be  used  as  money. 

UNITED  STATES  MONEY. 

242.  Tlie  un?  of  United  States  or  Federal  money  is  the 
Dollar. 

TABLE. 


10  Mills  (m.)  =  1  Cent    .    . 
10  Cents           =  1  Dime  .    . 
10  Dimes          =1  Dollar.     . 
10  Dollars        =  1  Eagle  .     . 

.    ct. 
.    d. 
.     $ 
.    E. 

Scale - 

$      d.        ct.         m, 
1  =  10  =  100  =  1000 
-Decimal. 

The  coins  of  the  United  States  are — 

Gold:  The  double-eagle,  eagle,  half-eagle,  quarter-eagle,  three- 
dollar  piece,  one-dollar  piece. 

Silver:  The  dollar,  half-dollar,  quarter-dollar,  the  twenty-cent 
piece,  the  ten-cent  piece. 

Nickel :    The  five-cent  piece  and  three-cent  piece. 

lironze  :   The  one-cent  piece. 

There  are  various  other  coins  of  the  United  States  in  circulation, 
but  they  are  not  coined  now. 


MEASURES   OF  VALUE.  159 

The  denominations  dimes  and  eagles  are  rarely  used,  the  dimes 
being  regarded  as  cents,  and  the  eagles  as  dollars. 

No  exampleii  in  Kcductioii  of  U.  S.  Money  are  given,  because  the 
pupil  has  been  familiarized  with  the  process  from  the  beginning. 

CANADA  MONEY. 

243.  The  currency  of  Canada  is  decimal,  and  the  table  and 
denoininatiom  are  the  same  as  those  of  United  States  money. 
English  money  is  still  used  to  some  extent. 

The  coins  of  Canada,  are,  for  the  most  part,  of  the  same  denomina- 
tions as  those  of  the  United  States,  except  the  gold  coins,  which  are 
the  sovereign  and  half-sovereign. 

ENGLISH  OR  STERLING  MONEY. 

.244.  English  money  is  the  currency  of  Great  Britain.    The 
unit  is  the  Pound  or  Sovereign. 


TABLE. 

4  Farthings  (far.)     =     1  Penny.    .     . 

d. 

12  Pence                      =     1  Shilling  .    . 

8. 

^OSLUHng,                ={1S;1- 

£ 

£      8.        d.        far. 

1=20  =  240  =  960 

Scale  — 4,  12,  20. 

1.  Farthings  are  commonly  written  as  fractions  of  a  |)enny. 
Thus,  7  pence  3  farthings  is  written  7|d.;  5  pence  1  farthing,  5Jd. 

2.  The  value  of  £]  or  sovereign  is  $4.8665  in  American  gold. 

The  coins  of  Great  Britain  in  general  use  are — 

Gold:  Sovereign,  half-sovereign,  and  guinea,  which  is  equal  to 
21  shillings. 

Silver:  The  crown  (equal  to 5 shillings),  half-crown,  florin  (equal 
to  2  shillings),  shilling,  six-iK'nny  and  tliret'-|)enny  pieces. 

Copper:    IVnny,  half-iKMUiy.  and  fnrtliinu:. 


1(50  DE.NO.Ml>ATE    MMBKIK. 


UEO L U IIU.N  DESOEiNDLN U. 

245.  1.  How  many  farthings  are  there  in  2  pciK  /  In  .> 
pence?     In  7  jxincc?     In  8  jx^nce?     In  0  pence? 

2.  How  many  pence  are  there  in  2  shillings?  In  5  shil- 
lings?   In  7  shillings?     In  8  shillings?     In  6  shillings? 

3.  How  many  pence  are  there  in  5s.?  In  5s.  aii«l  'M.  '  In 
78.  4d.?     In4s.  5d.?     In  6s.  8a.? 

4.  How  many  farthings  are  there  in  5d. ?  In  6d.  3  far. '!  In 
Sja.?     In  6.^.'?     In8Jd.?     In  OJd.?     In  lOJd.? 

5.  How  many  shillings  are  there  in  £2  5s.?     In  £3  5s.? 

240.  Reduction  of  a  denominate  number  is  the  process 
of  changing  it  from  one  denomination  to  another  without 
altering  its  value. 

247.  Keduetion  Descendlnf/  is  the  process  of  chang- 
ing a  denominate  number  to  an  equivalent  number  of  a  lower 
denomination. 

WKITTEN    JBXEBCI8B8, 

1.  How  many  farthings  are  there  in  £3  5s.  6fd.? 

PROCESS.  Analysis. — Since    in    1    pound 

no   f:„    aiKA  there  are  20  shillings,  in  3  pounds 

^y^        '      ^    '  there  are  3  times  20  shillings,  or  60 

shillings;  and  60  shillings  +  5  shil- 

65s.      =£3  58.  lings  =  65  shillings. 
12  Since  in  1  shilling  there  are  12 

rrog  J       __  £ o  - ^    g  J  pence,  in  65  shillings  there  are  65 

,  —        •  times  12  pence,  or  780  pence;  and 

780  pence  +  6  pence  =  786  pence. 


3147  far.  =  £3  5s.  6  Jd.  since  in  1  penny  there  are  4  far- 

things, in  786  pence  there  are  3144 
farthings;  and  3144  farthings  +  3  farthings  =  3147  farthings. 
Hence  in  £3  5s.  6|d.  there  are  3147  farthings. 


ruocEss. 

£f  =  ^  of  20s. 

=.Ys. 

iV^8.  =  ^of  12d. 

=  ^^d. 

f^d.  =  102fd. 

iM:i>r(  iioN   dj:sckm)IN(;.  161 

2.  Hou  iii;iii\  piiiic  lire  there  in  £2  10s.  G<J. ? 

.'i.  How  many  shillings  are  there  in  £13  5s.? 

4.  How  nuiny  farthings  arc  there  in  £4  6s.  5d.? 

f).  How  many  yxiuce  are  there  in  £^? 

Analysis.— Since  in  1  pound 
there  arc  20  Bhillings,  in  J  of  a 
pound  there  are  f  of  20  shil- 
lings, or  -^7°-  of  a  shilling. 

Since  in  1  shilling  there  are 
12  pence,  in  ^^  of   a  shilling 
there  are  ^^^  of  12  pence,  or  ^^  pence;  and  ^^  pence  =  102fd. 
Therefore  in  £}  there  are  102fd. 

Rule. — Multiply  the  number  of  the  highest  denomination  given, 
by  the  number  of  units  of  the  next  lower  denomination  which  is 
equal  to  one  of  the  next  higher,  and  to  the  product  add  tJie  num- 
ber given  ofUds  lowei'  denomination. 

Proceed  in  like  manner  with  tliis  and  each  successive  remit 
thus  obtainedf  until  the  number  is  reduced  to  tlie  denominatian 
required. 

6.  How  many  pence  are  there  in  £|? 

7.  How  many  pence  are  there  in  £J? 

8.  How  many  farthings  are  there  in  ^s.  ? 
0.  How  many  pence  are  there  in  £3^? 

10.  How  many  shillings  are  there  in  £5  6s.?  How  many 
farthings? 

1 1.  Reduce  128.  5d.  2  far.  to  farthings. 

12.  How  many  pence  are  there  in  £7  9s.  5d.  ? 
!.'>.  Reduce  178.  6}d.  to  farthings. 

1 4.  What  is  the  value  of  £J  in  units  of  lower  denomina- 
tions? 

15.  Find  the  number  of  farthings  in  £5  13s.  3tl. 

16.  Reduce  £35  6s.  8d.  to  pence. 

17.  Reduce  £45  3s.  9Jd.  to  farthings. 
IH.  Reduce  £29  188.  5<1.  to  farthings. 


1^)2  ni,\'  »M  1  N  \  1  I,   M  \i  r.i'.R.' 


REDIUTIU.N  ASCENDING. 

24S.  1.  How  many  pence  are  there  in  12  fartliinfrs?  Tn 
1()  farthings?     In  20  farthings? 

2.  How  many  shillings  are  there  in  '21  pence?  In  60 
pence?    84  pence?    96  pence? 

3.  How  many  pounds  arc  thrro  in  in  >!iilliiiL:<?  In  00 
shillings?     In  120  shillings? 

4.  How  many  pounds  sterling  must  be  paid  for  10  jmirs 
of  boots  at  6  shillings  a  pair? 

5.  At  5  shillings  each  how  many  ]X)unds  sterling  must  be 
paid  for  16  hats?    For  20  hats? 

6.  Sold  8  pairs  of  skates  at  5  shillings  a  pair.  How  many 
pounds  sterling  did  I  receive  for  them? 

Redaction  Asretidinf/  is  the  process  of  changing  a 
denominate  number  to  an  equivalent  number  of  a  higher 
denomination. 

WRITTEN    EXERCISES. 

1.  How  many  pounds  sterling  are  there  in  7254  pence? 

TROCESS.  Analysis. — Since  12  pence  arc 

12)7254  efiual  to  1   shilling,   there  must 

be    as   many   shillings    in    7254 

z  U  J  DU4  .  .  .  D  pence  as  12  pence  arc  contained 

30  ....  4  times  in  that  number.     12  pence 

rr  otzjA   ^QA    A  ft!          ^^^  contained  in  7254  j^ence  604 

7Zb4(i.=i.6i)    4  s.  bd.        times  with  a  remainder  of  6  i)ence, 

therefore  7254  j^ence  are  equal  to  604s.  6d. 

Since  20  shilling  are  equal  to  1  i^ound,  there  must  be  as  many 
pounds  in  604  shillings  as  20  shillings  are  contained  times  in  that 
number.  20  shillings  are  contained  in  004  shillings  30  times  and  a 
remainder  of  4  shillings. 

Therefore  7254  pence  are  equal  to  £30  4s.  6d. 


2.  How  nmny  sliillings  are  tliero  in  o4.")  ilirlliiiigsy 

3.  IIow  many  pounds  are  there  in  45G  shillings? 

4.  IIow  many  pounds  are  there  in  1586  pence? 

5.  Reduce  3864  flirthings  to  pounds. 

6.  Reduce  fd.  to  a  fraction  of  a  pound. 

PROCESS.  Analysis. — Since   1    penny  is 

1.1  —  i  of  J,Q  —   Vs  T^  ^"^  ""  Bhilling,  f  of  a  penny  is 

fd.—  f  oi  fjs.  —  -g^s,         ^^^^j^j  ^^  2  of  yV  of  a  shilling,  or 

^\B.  =  ^\0f£^~-=^£T^      A  of  a  shilling 

Since  1  shilling  is  75«iy  of  a 
pound,  ^  of  a  shilling  is  equal  lo  ^  of  2*5  of  a  pound,  or  -^5^^  of  a 
ix)und. 

Rule. — Divide  tlie  given  number  by  iJie  mimbei'  of  tlud  de- 
nomination which  is  equal  to  a  unit  of  Uie  7iext  higher  denomi- 
nation. 

'  Divide  the  quotient  in  like  manneVy  and  Hius  proceed  until  iJie 
required  denomination  is  reached. 

The  lad  quotient  and  the  several  remainders  ivill  be  Vie  result 
sought. 

7.  Change  ^  of  a  shilling  to  a  fraction  of  a  jwund. 

8.  Change  1  of  a  farthing  to  a  fraction  of  a  shilling. 

9.  Change  384  pence  to  units  of  higher  denominations. 

10.  Change  81  ITi  shillincr>  to  pounds. 

Reduce:  Reduce: 

11.  3596d.  to  \Y)uui\<.  20.  £15  83.  to  farthings. 

12.  3846  far.  to  shillings.  |  21.  £15  to  dollai-s. 

13.  4856s.  to  jwunds.  !  22.  $456  to  jxiunds. 

14.  5968  far.  to  pounds.  |  23.  $394.45  to  i)ounds. 


15.  3984d.  toix)unds. 

16.  4685  far.  to  shillings. 

17.  48567  far.  to  pounds. 

18.  £3  14s.  5d.  to  far. 

19.  4.H096  far.  to  pounds. 


24.  $37.50  to  pounds. 

25.  £25  to  dollars. 

26.  £15  lOs.  to  farthings. 

27.  $973.30  to  i)ounds. 

28.  $1216.625  to  iwunds. 


11)4  J'i..N<)MiNAl  i:  M MlJKiCS. 


FRENCH  MONEY. 

249,  In  Fmnoe  the  (mim-H'  v    i-  dtcLuud.     Tin-  ?/////  i-  the 

Franc. 

lAi>Li:. 

10  CentiniCH  (ct.)  [pronounced  8on-4eevis\  =  1  Dccinio     .     .    dc. 
10  DecimcH  [pronounced  (/es-Menu]  =  1  Franc  .     .     .     fr. 

Seak  —  Decimal. 

The  vahie  of  the  franc,  a«  detenu ined  by  the  Secretary  of  the  Treas- 
ury, IB  $.193  in  United  States  money. 

1.  How  maoy  centimes  are  there  in  1  franc?    In  5  francs? 

2.  How  many  decimes  are  there  in  1  franc?    In  7  francs? 

3.  How  many  centimes  are  there  in  4  decimes? 

4.  How  many  dollars  are  there  in  10  francs?  In  20  francs? 

5.  In  3684  centimes  how  many  francs  are  there? 

6.  How  many  franc-s  are  there  in  619.30?    In  89.65?    In 
S3. 86? 


MEASURES  OP  SPACE. 

250.  S[iace  is  extension   in  any  direction.     It  has  three 
dimensions  or  measurements — length,  breadth  and  thickness. 

251.  A  Line  is  that  which  has  only  length. 

Til  us,  the  edge  of  any  thing,  or  the  distance  between  any  two  objects 
or  places,  is  a  line. 

252.  A  Surface   is   that  which   has   only  length   and 
breadth. 

Thus,  the  floor,  this  page,  or  tlie  outside  of  any  thing,  is  a  surface. 

253.  A  Solid  is  that   which    has  length,   breadth   and 
thickness. 

Thus,  a  stone,  an  apple,  a  block,  a  book,  etc.,  are  solkh. 


MEAsi'RFs   or   SPACE. 


1G5 


LINEAR  MEASURES. 

254.  Linear  Measures  ai-e  used  in  measuring  lengths 
and  distances. 


LIXEAI 

\  MEASURE. 

SURVEYORS'  LINEAR  MEAS 

12  Inchcfl  (in 

0  =  1  Foot    . 

ft. 

7.92  Inches    =  1  Link  .    .    1. 

3  Feet 

=  1  Yard  . 

yd. 

2o  Links     =  1  Rod    .     .     rd. 

h\  Yards| 
\&\  Feet    j 

=  1  Rod     . 

rd. 

4  Rods  ),,,,.              , 
-./x/^T-   1     >-=l  Chain      .     eh. 
100  Links  j 

320  Rods 

=  1  Mile    . 

mi. 

80  Chains  =  1  Mile  .    .    mi. 

ml     rd. 

yd.         /I.           in. 

1=320  = 

:1760 

—  5280  —  63360 

&«/«-- 12,  3,  5i,  and  320. 

The  following  are  also  used : 

3  Barleycorns 

4  Inches 
6  Feet 
3  Feet 

5  Paces 


8  Furlongs  : 

1.15  Statute  Miles 
3  Geographical  Miles: 
60  Geographic  Miles") 


).I6  Statute  Miles 


1  Inch.    Used  by  shoemakers. 

1  Hand.   Usetl  to  measure  the  height  of  horses. 

1  Fathom.    Used  to  measure  depths  at  sea. 

1  TtnA  '  I    ^^  *"  pacing  distances. 

1  Mile. 

:  1  Greographical,  or  Nautical  Mile. 
1  League. 

1 


^^^^  fof  Latitude  on  a  Meridian,  or 
^^^\of  Longitude  on  the  £(piator. 


1.  For  the  purpose  of  moasuriog  cloth  and  other  goods  sold  by  the 
yard,  the  yard  is  divideil  into  halves,  fourths,  eighths  «nd  sixteenths. 

2.  The  length  of  a  degree  of  latitude  varies.     CD.IG  is  the  average 
length,  and  is  that  adoptetl  by  the  l^nitetl  States  Coast  Survey. 


166  DEXOMi\\ii:    M   Mi;r-:ns. 

1.  How  many  iiui.v.^   ..:-     ;>.-,.     ,.,     ,    ;■  ■  ;  .     i.   i.>t?     8 
feet?    10  feet?     12  feet? 

2.  How  many  feet  are  there  in  2  nnls?     3  rods?     4  nxls? 

3.  How  mnnv  ?»«"'>'•<  ;?»•"  tl^To  in  2  yards?     4  yard;*?     5 
yards? 

4.  How  many  inches  are  there  in  2  yards  and  2  inches? 
3  yards  and  4  inches? 

5.  How  many  rods  are  there  in  2  miles?     3  miles? 

6.  How  many  feet  are  there  in  1  rod  and  2  yards?    2 
rods  and  3  yanis? 

7.  How  many  feet  are  there  io  45  inches?    In  63  inches? 

8.  How  many  yards  are  in  22  feet?    In  47  feet?    In  34 
feet? 

9.  How  many  miles  in  640  rods?    In  480  rods? 

10.  How  many  inches  in  10  links?     In  100  links? 

11.  How  many  links  in  5  rods?     In  3  rods?     In  6  rods? 

12.  The  length  of  a  road  was  400  links.    What  was  its  length 
in  rods? 

13.  In  160  chains  how  manv  miles? 


WBITTSN    EXERCISES, 

14.  Reduce  5  mi.  18  rd.  4  yd.  to  yards. 

15.  Reduce  7  rd.  5  ft.  6  in.  to  inches. 

16.  How  many  inches  are  there  in  7  miles?     In  9  miles? 

17.  A  building  was  327  ft.  long.     How  many  rods  was  it 
in  length? 

18.  A  man  sold  a  piece  of  wire  36828  in.  long.     How 
many  rods  was  it  in  length? 

19.  In  3900  rods  how  many  miles  are  there? 

20.  Reduce  15  mi.  8  rd.  5  yd.  3  ft.  4  in.  to  inches. 

21.  Reduce  8  mi.  14  rd.  5  ft.  4  in.  to  inches. 

22.  Reduce  66454  niches  to  miles,  etc. 

23.  Reduce  158964  inches  to  miles,  etc. 


H£ASUR£S  OF  SPACE. 


167 


24.  The  diameter  of  the  earth  is  7912  miles.     How  many 
feet  is  it  ' 

25.  How  high  is  a  horse  that  measures  15  hands? 

2().  My  farm  is  67  ch.  83  1.  long.     How  many  rods  long 
is  it? 

27.  Reduce  59  ch.  75  1.  to  inches. 


SURFACE  MEASURES. 

255.  An  Angle  is  the  difference  in 
the  direction  of  two  lines  that  meet. 

25(>.  A  Square  is  a  figure  that 
has  four  equal  sides,  and  four  equal 
angles. 

"  A  square  inch  is  a  square  whose  side  is  one 
inch.  A  square  foot,  a  square  whose  side  is 
one  foot. 

The  angles  of  a  square  are  called  right  angles. 

257.  A  Rectanffle  is  a  figure  that 
has  four  straight  sides  and  four  equal 
angles. 

The  angles  of  a  rectangle  are  all  right  angles. 

258.  The  Area  or  extent  of  any 
surface  is  tlio  numlx^r  of  square  units  it 
contains. 

Thus,  if  a  rectangle  is  4  inches  long  and 
3  inches  wide  the  area  will  be  12  square 
inches. 

For  it  may  l>e  divided  into  4  rows,  euch 
containing  3  square  inches  or  untfo,  and  the 
entire  area  will  be  12  square  inches. 

The  metlxHl  of  computing  the  area  of  fig- 
ures that   are   not   rectangular   is  given    in 

MeN8U  RATION. 


-i 


168  DENOMIN'ATF    NT'MRERS. 

259.  The  '.:.:  '  f'>  /•'    ^,r,uh,rt  of  live 

numbers  Hud  expr- 

The  length  and  breadth  muni  be  expremed  in  tiniUi  of  the  Rame  de- 
notuinatioT^ 

1.  Ho\»     ill. Ill  >     .-«|uai\-    iinlu.T     aic      IIk-Ii-      iil    ii      ItAiuiiUi'-    ^» 

inch^  long  and  5  inches  wide?     8  inches  long  and  3  inches 
wide?    7  inches  long  and  5  inches  wide? 

2.  How  many  square  feet  are  there  in  a  rectangle  4  feet 

long  and  3  feet  wide?    7  feet  long  and  5  feet  wide? 

3.  How  many  square  inches  are  there  in  a  square  whose 

side  is  2  inches?    6  inches?    8  inches?     12  inches? 

4.  How  many  square  yanls  are  there  in  a  square  whose 

side  is  2  yards?    5  yards?    7  yards?    10  yards? 

5.  How  many  square  feet  are  there  in  a  square  whose  side 

is  I  yanl  long?    3  yanls?    5  yards?     7  yards?     10  yards? 

6.  How  many  square  rods  are  there  in  a  lot  5  rods  long 

and  4  rods  broad?    In  a  square  whose  side  is  6  rods? 

7.  How  many  square  feet  in  a  square  whose  side  is  3  yards? 
In  a  rectangle  whose  length  is  4  yards  and  breadth  3  yards? 

8.  How  many  square  inches  are  there  in  a  square  foot? 

Square  feet  in  a  square  yard?    Square  yanls  in  a  square  rod? 

SQUARE  MEASURE. 
TABLE. 


144  SqUcire  Inches  (sq.  in.)  =  1  Square  Foot    . 
9  iSqiiare  Feet  =  1  Square  Yard  . 

30}  Square  Yards  =  1  Square  Rod     . 

160  Stjuare  Rods  =  1  Acre       .    .    . 

640  Acres  =  1  Square  Mile    . 


sq.  ft. 
sq.  yd. 
sq.  ni. 
A. 
sq.  mi. 


aq.mi,  A.         sq.nL         sq.  yfl.  f^i-ff.  sq.in. 

1  =  640  =  102400  =  3097600  =  27878400  =  401 4489600 


&a/c— 144,  0,  30i,  160,  640. 


MEASURES  OF  SPACE.  169 

I.  Plastering,  ceiling,  etc.,  are  commonly  estimated  by  the  ttquare 
yard;  paving,  glazing,  and  ntone-ciitting,  by  the  sfjnare  fool. 

2-  Rooting,  flooring  and  slating  are  commonly  estimated  by  the 
fuputre  of  100  /ee/. 

SURVEYORS'  SQUARE  MEASURE. 

TABLE. 

625  Square  Linkn    =1  «q.  rd.        I     10  Square  Chains       =1  acre. 
16  Square  Rods    =1  sq.  chain.  |  640  Acres  =-1  sq.  mi. 

In  some  parts  of  the  country  a  Township  contains  ?,(\  «<]unre  miles, 
or  is  6  miles  square. 

1.  How  many  square  feet  are  there  in  4  square  yards? 
7  square  yards?    9  square  yards? 

2.  How  many  square  inches  are  there  in  2  square  feet? 
3  square  feet?     5  square  feet? 

3.  How  many  square  yards  are  there  in  27  square  feet? 
36  square  feet?    81  square  feet? 

4.  How  many  square  yards  are  there  in  10  square  rods? 

5.  How  many  square  chains  are  there  in  48  square  rods? 
64  square  rods?    96  square  rods? 

6.  How  many  square  rods  are  there  in  3  acres?     In  5 
acres? 

7.  How  many  acres  are  there  in  480  square  rods? 

8.  How  many  square  feet  are  there  in  288  square  inches? 

9.  How  many  acres  are  there  in  30  square  chains?     In 
60? 

WJtTTTi:N     EXERCISES. 

10.  Reduce  9  sq.  yd.  i>  s<].  ft.  15  sq.  in.  to  square  inches. 

II.  Reduce  3  s(j.  mi.  15  st].  rd.  to  square  inches. 

12.  R^^-duce  262685  sq.  ft.  to  acres,  etc. 

13.  Reduce  2  A.  37  sq.  n\.  5  sq.  yd.  7  sq.  ft.  to  sq.  in. 

14.  Reduce  184265  scj.  in.  to  uniu*  of  higher  denominations. 


170  DFTNOMINATE   NUMBERS. 

15.  Reduce  -f  of  an  acre  to  units  of  lower  denominatioDs. 

PROCESS. 

^  A.  X160.^A^flq.rd.        =114f8q.rd. 
f  bq.  rd.  X  30 J  =  f  sq.  rd.  X  H^  =  W^Jq.  yd.  =  8t\  sq.  yd. 
T»f8q.yd.X     9  =  n8q.{t.  =5H8q.ft. 

lieq.ft   Xl44  =  -4|^Bq.in.       ^llSffiq.in. 

Therefore  f  A.  =  114  sq.  rd.  8  aq.  yd.  5  Bq.  ft.  113J  sq.  in. 

Analysis. — We  multiply  by  that  number  in  the  Rcale  which  will 
reduce  the  number  to  the  next  lower  denomination,  and  so  continue 
to  multiply  each  fraction  until  the  lowest  denomination  is  reached. 

16.  Express  -J  of  an  acre  in  lower  denominations. 

17.  What  iK'ii  >  i'  ail  acre  an  KM)  m,.  nl.'.^  ^')  hj.  rd.? 
120  sq.  rd.? 

18.  Change  f  of  a  sq.  rd.  to  lower  denominations. 

19.  How  many  sq.  in.  are  there  in  a  rectangle  7  inches 
wide  by  11  inches  long? 

20.  How  many  square  feet  are  there  in  a  floor  8  feet  long  i^ 
by  15  feet  wide? 

21.  How  many  square  yards  are  tlicre  in  a  ceiling  that  is    /  . 
18  feet  wide  by  21  feet  long?  w^|^  :_.r.    ijl 

22.  What  is  the  area  of  a  square  whose  side  is  5  feet?       |  ^'j 

23.  How  many  square  yards  are  there  in  a  floor  18  feet      ' 
wide  by  24  feet  long?     How  much  would  it  cost  to  carpet 

it  at  $1.15  per  square  yard? 

24.  How  many  yards  of  carpeting  1  yard  wide  will  be 
required  to  cover  a  room  18  ft.  long  by  17  ft.  wide? 

25.  What  will  it  cost  to  carpet  a  room  18  ft.  long  by  15 J 
ft.  wide,  with  carpet  f  of  a  yard  wide,  at  81.90  per  yard? 

26.  If  the  width  of  a  lot  is  66  feet,  how  long  must  it  be 
to  contain  \  of  an  acre?  What  will  be  the  cost  of  it  at 
33.25  per  square  foot? 

27.  A  pasture  containing  10  acres  had  a  width  of  20  rods?     U 
How  long  was  it? 


MEASURES  OF  VOLLME. 


171 


28.  Mr.  A.  solil  a  lot  of  land  whose  width  was  20  rd. 
and  whose  lengtli  was  80  rd.  at  S47.25  jxir  acre.  How 
much  did  he  get  for  it? 

29.  What  is  the  difference  between  10  square  feet  and  10 
feet  square?    lUustrate  this  by  drawings. 

30.  What  will  be  the  expense  of  painting  a  roof  48  feet      . 
long  and  22  feet  wide  at  $.30  a  square  yard?  %^9^^ 

31.  What  will  be  the  cost  of  cementing  the  bottom  of  a 
xjellar  45  feet  by  32  feet  at  §.30  per  square  yard?     pi4-t 

32.  How  many  yards  of  plastering  are  there  in  the  sides 
of  a  room  18  ft.  long,  17  ft.  wide,  and  11  ft.  high?  How 
many  in  the  ceiling?  What  will  l)e  the  cost  of  plastering  at 
$.37  a  square  yard? 

33.  What  will  be  the  cost  of  papering  the  side  walls  of 
the  above  room  at  3.25  per  square  yard? 


MEASURES  OF  VOLUME. 


2G().  A  Solid  has  length,  breadth,  iin<l  tli 


261.  A  Cube  is  a  solid  having 
six  equal  square  sides  called  faces. 

262.  A  Cubic  Inch  is  a 

solid  whose  sides  or  faces  are  each 
a  square  inch. 

263.  A  Cubic  Foot  h&soWd 

whose  sides  are  each  a  square  foot. 

264.  The  Volume,  or  Sol  if' 
Contents,  of  any  body  is  tli< 
number  of  solid  units  it  contain> 

Thus,  if  a  m\u\  in  4  ft   lonpr,  3  f 
wide,  and  3  ft.  thick,  its  vohnuc  wili 


I  loss. 


\^^ 


y^l^ 


:Tr 


1  i  '1  DENOMINATE    NUMBERS. 

be  36  cubic  feet.  For  it  may  l)e  divided  into  3  blocks,  oacli  conlnin- 
ing  12  cubic  feet,  ranking  in  all  3G  cubic  feet.  That  is,  the  nunilxr 
of  cubic  feet  in  each  block  will  be  equal  to  the  protluct  of  the  num- 
bers expressing  its  length  and  breadth,  and  the  number  of  blocks  is 
(Miual  to  the  number  expressing  the  thickness.     Therefore, 

2G5.  The  volume  o/*  any  redatiffular  solid  in  equal  to  the  prod- 
uct of  the  ntwibers  expressing  its  lengtii,  breadth^  and  thickneits. 

The  length,  breadth  and  thickness  must  be  expressed  in  units  of 
the  same  denomination. 

1.  How  many  cubic  feet  are  there  in  u  rectangular  solid 
whose  length  is  3  fl.,*its  b^adth  2  ft.,  and  its  thickness  2  fl.? 

2.  How  many  cubic  feet  are  there  in  a  cube  whose  dimen- 
sions are  each  3  feet;  or,  how  many  cubic  feet  are  there  in 
a  cubic  yanl  ?     In  a  cube  whose  sides  are  5  ft.  long  ? 

3.  How  many  cubic  inches  are  there  in  a  cube  whose  di- 
mensions are  each  12  inches;  or,  how  many  cubic  inches  are 
there  in  a  cubic  foot?    In  a  cube  whose  sides  are  10  in.  long? 

4.  AVhat  is  the  volume  of  a  cube  whose  sides  are  each  4 
inches  square?     9  inches  square?     16  inches  square? 


CUBIC  IMEASURE. 

TABLE. 

1728  Cubic  Inches  (cu.  in.)  =  1  Cubic  Foot     .     .     .    cu.  ft. 
27  Cubic  Feet  =  1  Cubic  Yard    .     .     .    cu.  yd. 

A  cord  of  wood  or 
stone  is  a  pile  8  feet 
long,  4  feet  wide  and  4 
feet  high. 

A  pile  that  is  1  foot 
long,  4  feet  wide  and  4 
feet  high,  is  a  cord  foot. 


MKA&Lilii->    «)1'     Vol.IMi;.  17? 

The  following  arc  the  denoiniiiatioii.'^: 

10  Cubic  Feet    =1  Cord  Foot     .     .     .     cd.  ft. 
8  Cord    Feet)  _  ,  ^     , 
128  Cubic  Feet  j"~  

1.  A  jyeirh  of  stone  or  masonry  is  16\  ft.  long,  1\  ft.  thick,  and  1  foot 
liigh,  and  contains  24J  eu.  ft. . 

2.  A  cubic  yard  of  earth  is  considered  a  locid. 

3.  Brick-work  is  commonly  estimated  by  the  thousand  bricks. 

4.  Brick-layers,  masons  and  joiners  commonly  make  a  deduction 
of  one-half  the  space  occupied  by  windows  and  doors  in  the  walls  of 
buildings. 

5.  In  computing  the  contents  of  walls,  masons  and  brick-layers  mul- 
tiply the  entire  distance  around  on  the  outside  of  the  wall  by  tiie 
height  and  thickness.    The  comers  are  thus  measured  twice. 


WniTTEy     EXERCISES. 

1.  How  many  cubic  inches  are  there  in  2  cubic  feet?    In 
3cu.  ft.?    In  15  cu.  ft.?    In  32  cu.  ft.? 

2.  How  many  cubic  feet  are  there  in  2  cubic  yards?     In 
3  cu.  yd.?     In  13  cu.  yd.?     In  25  cu.  yd.? 

3.  How  many  cubic  feet  are  there  in  5  cords?    In  8  cords? 

4.  How  many  perch  of  masonry  are  tliere  in  418  cubic  feet? 
What  will  Ixi  the  cost  of  laying  it  at  31.75  per  perch? 

5.  How  many  perch  of  masonry  are  there  in  a  wall  38  feet 
long,  4  feet  high,  and  H  feet  thick? 

(>.  How  many  yards  or  loads,  of  earth,  must  be  removed 
in  digging  a  cellar  35  feet  by  20,  8  feet  deep? 

7.  Reduce  32  cu.  ft,  1 14  cu.  in.  to  cubic  inches. 

8.  Reduce  13  cu.  yd.  18  cu.  fl.  to  cubic  feet. 

9.  Rwluce  15  perch  13J  cu.  ft.  to  cubic  feet. 

10.  How  many  cubic  blocks  of  one  foot  on  a  si(b   <  an  in' 
cut  from  a  cuIk^  that  is  8  yards  long  on  each  edge? 

11.  How  n)any  cubic  feet  in  a  block  of  marble  9  feet  long, 
5  feet  wide,  and  3.t  feet  thick?        ,  i^  *,   / 


1  74  DENOMINATE^  NUMBERS. 

12.  A  man  sawed  a  pile  of  woocl  40  ft.  long,  4  ft.  wide,  and    Ij^ 
h\  ft.  high,  for  81.50  jxir  cord.     How  much  did  he  earn?      ^^ 

13.  A  bin  is  8  fl.  long,  7  ft.  wide,  and  5  ft.  high.  How 
nuiny  cubic  feet  are  there  in  it?  How  many  cubic  inches? 
How  many  bushels  will  it  hold  if  a  bushel  contains  2150.4 
cubic  inches? 

14.  What  will  it  cost  to  excavate  a  cellar  80  by  35  ft.,  and 
8  ft.  deep,  at  8 .42  |)or  \A.  ?  What  will  l)e  tlie  expense  of  build- 
ing a  stone  wall  around  it  \\  ft.  thick,  at  $3.75  a  {)erch? 

15.  How  many  bricks  will  it  require  to  build  a  wall  35J- 
ft.  long,  19  ft.  high,  and  3  ft.  thick,  allowing;  'I'l  liricks  to 
the  cubic  foot  when  laid  ? 


BOARD  MEASURE. 

2G6.  In  measuring  lumber,  when  a  lyiard  is  one  hich  ihickj 
the  numl)er  of  feet  board  measure  is  obtained  by  multiplying 
the  length  in  feet  by  the  brendth  expressed  in  feet. 

When  the  lumber  is  more  than  one  incJi  thicky  tlie  numl3er 
of  feet,  board  measure,  may  be  obtained  by  nmltiplying  the 
length  in  feet  by  the  breadth  in  feet,  and  this  product  by  the 
number  expressing  the  inches  in  thickness. 

When  a  Ixjard  tapers  uniforndy, 
the  average  or  mean  width  is 
equal  to  half  the  sum  of  the  two 
ends. 

Board  measure  may  also  be  computed  by  multiplying  the  number 
of  feet  in  length  by  the  number  of  inches  in  width,  and  then  dividing 
the  product  by  12. 

BXHJtCISES. 

How  many  feet  are  there  in  the  following  boards: 


1.  18  ft.  by  16  in.? 

2.  15  ft.  by  11  in.? 


3.  10  ft.  by  13  in.? 

4.  13  ft.  by  15  in.? 


5.  How  many  feet  of  timber  are  tliere  in  a  stick  40  feet^J^/ 
long,  9  inches  wide,  and  (5  inches  thick? 

G.  Mr.  B.  bonght  318  fence  l)oards  IG  feet  long  and  8  inches 
wide.     What  did  they  cost  at  811  per  thousand  feet? 

7.  A  lumber  dealer  lx)ught  35  three-inch  planks,  22  feet 
long  and  16  inches  wide,  at  $17.50  per  M.  How  much  did 
they  cost? 

s.  What  will  it  cost  to  floor  a  room  35  feet  by  18,  with  IJ 
inch  flooring,  at  630  \m  M,  allowing  \  for  matching? 

9.  What  will  be  the  expense  of  flooring  a  room  20  feet 
by  25  with  \\  iuch  flooring,  at  $25  per  M,  allowing  \  for 
matching? 


MEASURES   OP  CAPACITY. 

LIQUID  MEASURE. 
267.  Liqnid  Jleasurc  is  used  in  measuring  liquids* 

TABLE. 

4  Gill*  (gi.)  =  l  Pint         .     .     .    pt. 
2  Pints  ~1  Quart       .     .     .     qt. 

4  QiKirfu       -    1  Gallon     .     .     .     i»al, 

'.        '//.        lA.         <ii. 

1  =  4  =  8  =  32 
Scale-  1.  ::,   1. 

1.  In  determining  the  capacity  of  eistemSj  reservoirSj  etc.,  3U  gallona 
are  conKJdercd  a  barrel  (bbl.),  and  2  barrels,  or  63  gallons  a  hogshead 
(ldj<l.).  In  commerce,  however,  the  barrel  and  hogshead  are  not  fixed 
measures. 

2.  Canks  of  large  sixe  do  not  hold  any  fixed  quantity.  Their  ca- 
pacity U  usually  market!  upon  them. 

3.  The  standard  (jidhn  of  the  United  States  contains  231  cubic 
niches. 

4.  The  beer  (jidlon  is  not  now  in  use.     It  <  ontaint  d  282  cuhif  inches. 


17G  M'^'  '^'  '  ^  ^  If     \  I    M  !'!   i: 


EXERCISES. 


1.  How  many  gills  are  there  in  3  pints?  5  pints?   7  pints? 

2.  How  many  gills  are  there  in  2  quarts?     3  quarts? 

3.  How  many  pints  are  there  in  3  quarts?     8  quarts? 

4.  How  many  pints  are  there  in  a  cask  which  contains 
37  gallons? 

5.  A  man  soKl  ()^^l  pints  of  milk  at  20  cents  a  gallon. 
How  much  did  he  get  for  it?    How  many  gallons  were  there? 

().  lieduce  3846  gi.  to  gal.     4869  pt.  to  gal. 

7.  Ileducc  3  gal.  4  qt.  1  pt.  3  gi.  to  gi. 

8.  Reduce  4  bbl.  6  gal.  to  gi.     484  pt.  to  gal. 

9.  Reduce  24  gal.  to  pt.     8459  gi.  to  bbl. 

10.  How  many  cubic  inches  are  there  in  7  gal.? 

1 1 .  How  many  gallons   will  a  vessel   hold   that  contains 
3846  cubic  inches? 

12.  How  many  barrels  of  water  will  a  cistern  hold  that  is 
J5  feet  long,  10  feet  wide,  and  8  feet  deep  ? 


APOTHECARIES'  LIQUID  MEASURE. 

268.  Apothecaries^  Liquid  Pleasure  is  used  in 
com|)ounding  and  measuring  liquid  medicines. 

TABLE. 

60  Drops  (gtt.)  or  minims  (TT[)  =  1  Fluid  drachm  .  f^. 

8  Fluid  drachms  =  1  Fluid  ounce     .  /g. 

]G  Fluid  ounces  =  1  Pint         .     .     .  O. 

8  Pints  =  1  Gallon     .     .     .  CtJig. 

1.  The  abbreviation  Qmff.  is  from  the  Latin  congius,  a  gallon.  A 
jiint  being  one-eighth  cf  a  gallon  the  abbreviation  is  0.,  from  the  Latin 
«tcfuvus,  one-eighth. 

2.  In  writing  prescriptions,  physicians  wrlie  the  number  after  the 
Kymlx>l;  thus:  0.  5,  /^  2,  etc. 


M'AOITV.  177 


DRY   MEASURE. 
269.  Dvy  Measure  is  used  in  measuring  grain,  roots, 

Iruit,  etc. 

TABLK. 

2  Pinte  (pt.)  =  1  Quai  t  .  .  .  qt. 
8  Quarts  =  1  Peck  .  .  .  pk. 
4  Pecks  =  1  Bushel    .     .     .     bu. 

frtt.      "pk.       qt.        pf. 
1  =  4  =  32  =  64 
Scale— 2,  8,  4. 

1.  In  measuring  grain,  seeds,  or  small  fruits,  the  measure  must  be 
even  full  or  stricken.  In  measuring  large  fruits,  coai-se  vegetables,  corn 
in  the  ear,  ett\,  the  measure  should  be  heaped  at  least  six  inches. 

2.  Five  stricken  bushels  are  considered  equal  to  4  heaj)cd  bu.shels. 

3.  A  standard  bushel  contains  2150.4  cubic  inches. 

4.  A  pint,  quart,  or  gallon,  dry  measure,  is  more  than  the  same' 
quantity  liquid  measure,  for  a  quart  is  ^V  of  a  bushel,  or  ^  of  2150.4 
rubic  inches,  which  is  about  67^  cubic  inches,  while  a  quart  liquid 
measure  is  \  of  231  cubic  inches,  or  57|  cubic  inches. 

Cn.In.iii  Cn.In.iii  Cn.In.iii  ('ii.Tn.in 

OnoUHl.  OnoQt.  One  I't.  OneGi. 

Liquid  Mccis.    231  57|  28  J  7/, 

IhyMeaH.         268^  67^  33J  8| 


EXEnCJSES. 

1.  How  many  pints  are  there  in  3  quarts?    7  quarts? 

2.  How  many   qiuirts   nro   thoro   in   2   ]>opks?      *>    ]Hvk«? 
pecks?    7  j)ecks? 

3.  How  many  pints  are  there  in   1   bushel?     3  bushels? 
luis^hel.^?     8  bushels? 

4.  How  many  pints  are  there  in  3  bu.  3  pk.  5  qt  1  pt  ? 

5.  How  many  pints  are  there  in  8  bu.  5  qt.  3  pt  ? 

6.  Change  10^45  qt.  to  units  ot*  higher  den<miinutions. 

12 


178  DENOMINATE    NUMBERS. 

7.  Change  13965  pt.  to  iiniU  ui'  liighcr  ilenoininations. 

8.  Change  57364  qt.  to  units  of  higher  denominations. 

9.  Change  35  bu.  3  pk.  6  qt.  1  pt.  to  j)ints. 

10.  How  luanv  ciiImc  indus  are  there  in  7  bu.?     ("5  bu. ? 
10  bu.?    20  bu. 

11.  How  many  bushels  uic  there  in  13846  cu.  in.?     35769 
cu.  m.  ?     48695  cu.  in. 

12.  How  many  cubic  inches  are  there  in  a  bin  8  ft.  long, 
7  ft.  wide,  and  5  ft.  high  ?     How  many  bushels  will  it  hold  ? 

13.  How  many  bushels  will  a  bin  hold  that  is  9  ft.  long, 
6  ft.  wide,  and  6  ft.  high? 


MEASURES  OF  WEIGHT. 

270.  Wi^hjlit  i.-  \\\v  measure  of  the  force  tliat  attracts 
bodies  to  the  earth. 

A\'OtRDUP0IS   WEIGHT. 

271.  ArotrdHpols  WeUfht  is  used  in  measuring  all 
coarse  and  heavy  articles,  as  hay,  grain,  groceries,  coal,  etc., 
and  the  metals,  except  gold  and  silver. 

TABLE. 

16  Ounces  (oz.)  =  1  Pound lb. 

100  Pounds                  =  1  Hundred-weiurlit  cwt. 

20  Hundred-weight  =  1  Ton T. 


T.    act.       fb. 

oe. 

1  =  20 -=2000- 

^32000 

16, 100,  20. 

Scule- 

1.  In  weighing  coal  at  the  mines  and  in  levying  duties  at  the  United 
States  Custom  House,  the  Ivng  ton  of  2240  lb.  is  sometimes  used. 

2.  The  ounce  is  eonsideretl  as  10  drama. 


MKAr>l  iiK> 


179 


The  following  denominations  are  also  used 


56  lb.  Butter 
100  lb.  Grain  or  Flour 
100  lb.  Dried  Fish 
100  lb.  Nails 
196  lb.  Flour 
200  lb.  Pork  or  Beef 


=  1  Firkin. 
=  1  Cental. 
=  1  Quintal. 
=  1  Keg. 
=  1  Barrel. 
=  1  Barrel. 


280  lb.  Salt  at  N.  Y.  Works  =  1  Barrel. 

The  following  are  the  pounds  in  a  bushel  in  the  States 
named : 


Wheat, 

Indian  Cora... 

Oats 

liirley 

Buckwheats. . 

Rye 

<  lover  Seed.. 
Timothy  Seed 


^^li 


^l/TAll! 


G0|56|60j60jC0j60 

5250  56  52  56  56 


56, 


.1 


56 


1^ 


50  50  >Aj 
GO 


EXEUCI8E8. 


1 .  How  many  ounces  are  there  in  5  lb.  ?     Li  3  lb.  5  oz.  ? 

2.  How  many  jxiunds  are  there  in  5  cwt.  ?     In  6  cwt.  ? 
.*).  How  many  poiuids  are  there  in  1  ton  ?     In  3  T.  ? 

4.  How  many  [)ounds  are  there  in  3  T.  2  cwt.  5  lb.  ? 

5.  How  many  pounds  are  there  in  5  T.  216  lb.  ? 

6.  How  much  will  5  II).  7     /.     .f  indigo  cost  at   $.12^ 

IKT   oz.  ? 

7.  What  will  3J  lb.  of  confectionery  cost  at  $  .04^  per  oz.? 
«.  At  8  cents  a  pound,  what  must  be  jmid   for  5  cwt. 

28  lb.  of  sugar? 


180  |>I    \i  .\!  I  \   \   !  I       \l    MI'.KRS. 

9.    IIuW    lii;i»iv    |)uimu>  ;iii-   iiitu-   iti    i   IwUlt'l  ol"  pOrk?      Id 

J  barrel  of  siilt?     In  J  bi\rrel  of  flour?     In  \  keg  of  nails? 

10.  What  will  be  the  value  of  \  barrel  of  flour  at  68.50 
per  cwt.  ? 

11.  What  will  ^  quintal  of  codfish  cost  at  $.06^  per  lb.? 

12.  What  will  be  the  cost  of  13  cwt.  18  lb.  of  hay  at 
$15  ijcr  ton? 

1.3.  When  flour  is  $10  a  Imrrel,  how  many  pounds  can  I 
buy  for  $2.80? 

14.  A  merchant  soKl  o  cwt.  VJ  lb.  U  u/..  ol' clicc^c  at  S.17 
j)er  lb.     How  much  did  he  receive  for  it? 

15.  If  a  merchant  buys  flour  at  $9  jxjr  barrel  and  sells  it 
at  $5  per  cental,  how  luurh  will  Ik?  his  profit  on  the  sale  of 
15  barrels? 

16.  How  many  i>jirrels  of  salt  are  there  in  275000  lb.? 

17.  If  the  weight  of  a  bushel  of  wheat  Is  GO  lb.,  how  many 
bags  that  hold  2  bu.  each  will  be  required  to  carry  away  3  T. 
4  cwt.  20  lb.  of  wheat? 

TROY  WEIGHT. 

272.  Tro9/  Weight  is  used  in  weighing  gold,  silver, 

and  jewels. 

TABLE. 

24  Graias  (gr.)     =  1  Penny weiglit     .     .         pwt. 

20  Pennyweights  =  1  Ounce oz. 

12  ounces  =  1  Pound lb. 

/6.     oz.      put.        gr. 
1  =  12  =  240  =  5760 
.^(fe  — 24,  20,  12. 

1.  In  weighing  diamonds,  pearls,  and  other  jewels,  the  unit  com- 
monly eniployed  is  the  wml,  wliich  is  equal  to  4  grains. 

2.  The  term  camt  is  also  used  to  express  the  tineness  of  gold,  and 
means  r}^  part.  Thus,  gold  that  is  18  carats  fine  is  Jf  gold  and  /j 
cllov. 


MEASURES   OF    WKiCiilT.  181 


APOTHECARIES'  WEIGHT. 

273.  ApOtheeavies*  WeiyJtt  is  used  by  apothecaries 
!iud  physicians  in  weighing  medicines. 

TABLE. 

20  Grains  (gr.)  =  1  Scruple    .     .     .  sc.,  or  9 

3  Scruples        =  1  Dram  ....  dr.,  or  5 

8  Drams  =  1  Ounce  ....  oz.,  or  3 

12  Ounces  =  1  Pound  ....  lb.,  or  lb 

/6.      oz.      dr.       8C.        gr. 
1  =- 12  =  96  =  288  =  5760 

&a/c— 20,  3,  8,  12. 

1.  fn  writing  prescriptions,  physicians  express  the  number  in  Ro- 
man characters,  using  j  instead  of  i  final.  They  also  write  the  symbol 
first;   thus:  5V,  5vj,  ^ij. 

2.  Me<licines  arc  bought  and  sold  in  large  quantities  by  Avoirdu- 
pois Weight. 

1  lb.  Avoirdupois  =  7000  gr.     1  lb.  IT*"®^,  ""^  .    X  =  5760  gr. 
*  **  (Apothecaries')  ** 

1  oz.  "  =  437  J  gr.     1  oz.  "  =  480  gr. 


BX  ERC18E8. 

1.  How  many  j^rains  arc  there  in  3  pwt.?     In  5  pwt.  ? 

2.  How  many  |)onny  weights  are  there  in  5  oz.  ?    In  7  oz.  ? 

3.  How  many  grains  are  there  in  7  oz.  5  pwt  18  gr.  ? 

4.  Express  3456  grains  Troy  in  higher  units. 

5.  What  will  Ixj  the  value  of  an  ornaniont  weighing  2  oz. 
15  pwt.,  at  31.35  jKjr  pwt.? 

6.  How  many  spoons,  wei^^iiiiip   »  "..n  .  -  rai  li.  ran  be  made 
from  3  lb.  5  oz.  of  silver? 

7.  How  many  powders,  of  5  grains  each,  can  be  made  from 
5  oz.  7  dr.  of  quinine  ? 


182  DENOMINATE  NUMBEKS. 

MEASURES  OP  TIME. 

274.  Tlic  ibiiowing  are  the  ordinary  divisions  of  time : 
TABLE. 


60  Seconds  (sec., 

1  =  1  Minnio 

mm 

60  Minutes 

=  1  H<. 

.     hr. 

24  hours 

=  1  Day     . 

.     <la. 

7  days 

=  1  Week  . 

.     wk. 

365  days 

=  1  Year   .    . 

•   yr. 

366  days 

=  1  Leap  Year 

yr. 

100  years 

=  1  Century  . 

.    cen. 

yr.     mo.      da.        hr.  min.  sec. 

1  =  12  =  365  ==  8760  =  525600  =  31536000 

&«/€— 60,  60,  24,  365,  100. 

1.  In  most  business  computations  30  days  are  considered  a  month, 
and  12  months  a  year.    For  many  purposes  4  weeks  constitute  a  month. 

2.  The  common  year  contains  52  weeks  and  1  day,  the  leap  year  52 
weeks  and  2  days.  Hence,  commonly,  each  year  begins  one  day  later 
in  the  week,  but  the  year  succeeding  leap  year  begins  tuo  days  later. 

3.  The  time  required  for  the  earth  to  revolve  around  the  sun  is  one 
year,  which  is  365  da.  5  hr.  48  min.  49.7  sec.,  or  very  nearly  365|  days. 
Instead  of  reckoning  this  part  of  a  day  each  year,  it  is  disregarded,  and 
an  addition  made  when  this  would  amount  to  one  day,  which  would 
be  very  nearly  every  fourth  year.  This  addition  of  one  day  is  made 
to  the  month  of  February.  Since  the  part  of  a  day  that  is  disregarded 
when  365  days  are  considered  as  a  year,  is  a  little  less  than  oiie-quarler 
of  a  day,  the  addition  of  one  day  every  fourth  year  is  a  little  too  much, 
and,  to  correct  this  excess,  addition  is  made  to  only  every  fourth  cen- 
tennial year.  With  this  correction  the  error  does  not  amount  to  much 
more  than  a  day  in  4000  years.    Therefore, 

Centennial  years  whose  number  is  exactly  divisible  by  400, 
and  other  years  whose  number  is  exactly  divisible  by  4,  are 
Leap  Years, 


MEASURES  OF  TIME. 


183 


The  year  begins  with  the  month  of  January,  and  ends  with 
the  month  of  December. 

The  months,  their  names  and  the  number  of  days  in  each, 
are  as  follows: 


January, 

31  da.     .     .    Jan. 

July, 

31  da. 

.    July. 

February, 

28  or  29  da,    Feb. 

August, 

31  da. 

•    Aug. 

March, 

31  da.    .    .     Mar. 

September, 

30  da. 

.     Sept. 

April, 

soda.     .     .    Apr. 

October, 

31  da. 

.     Oct. 

May, 

31  da.    .     .    May. 

November, 

30  da. 

.    Nov. 

June, 

30  da.     .     .    June. 

December, 

31  da. 

.    Dec. 

' 

BXEn 

GISHS, 

1.  How  many  seconds  are  there  in  5  minutes?    In  6  min.  ? 

2.  How  many  minutes  are  there  in  \  hour?    In  \  hr.  ? 

3.  How  many  days  are  there  in  4  weeks?    In  5  wk.?    In 
8  wk.?     In  10  wk.? 

4.  How  many  hours  are  there  in  \  day?    In  \  da.? 

6.  How  many  days  are  there  in  \  year?     In  \  month? 

6.  What  part  of  an  hour  are  30  minutes?     15  min.? 

7.  How  many  hours  are  there  in  90  minutes?     In  120 
min.?     In  240  min.? 

8.  How  many  seconds  are  there  in  5  hr.  15  min.  12  sec.  ? 

9.  How  many  seconds  are  there  in  6  hr.  27  min.  38  sec.  ? 

10.  Express  in  units  of  higher  orders  48695  sec. 

11.  E.\piess  in  units  of  higher  orders  38497  sec. 

12.  How  many  minutes  are  there  in  5  yr.  of  365  da.  each? 

13.  How  many  days  are  there  from  Jan.  1st  to  May  Ist? 

14.  How  many  days  are  there  from  April  1st  to  Oct.  15th? 

15.  KecJiicc  2  wk.  5  da.  13  hr.  to  hours. 

16.  Reduce  5  da.  10  hr.  15  min.  to  minutes. 

17.  Retluce  384600  sec.  to  higher  denominations. 
IH.   Reduce  15  hr.  12  min.  18  .sec.  to  seconds. 
19.  Reduce  32965  min.  to  higher  denominations. 


181 


DENOMINATE  NUMBERS. 


CIRCULAR  OR  ANGULAR  MEASURE. 

275.  A  Circle  is  a  plaue  surface,  bounded  by  a  curved 

line  every  jwint  of  which  is 
equally  distant  from  a  jwint 
within  called  the  Center. 

276.  Tlie    Circum/'er' 

Cfice  is  the  line  that  bounds 
the  circle. 

277.  An  Arc  of  a  circle  is 
any  part  of  the  circumference. 

278.  A  Degree  is  -^  of 

the  circumference  of  a  circle. 

279.  The  Measure  of  an  Angle  is  that  part  of  the 
circumference  which  is  included  between  the  lines  which  form 
the  angle. 

Each  of  the  arcs  of  the  circumferences  ah,  cd,  DE,  is  a  measure 
of  the  same  angle,  and  therefore  contains  the  same  number  of  degrees  ; 
but  since  each  degree  is  -^  of  the  circumference,  the  length  of  a  de- 
gree must  vary. 

280.  Circular  or  Angular  Measure  is  used  to 
measure  arcs  of  circles  and  angles,  in  determining  latitude, 
longitude,  direction,  the  position  of  vessels  at  sea,  etc. 

TABLE. 


60  Seconds  ( ^^ )  =  1  Minute    .     .     . 

60  Minutes         =  1  Degree     .     .     . 

360  Degrees  =  1  Circumference  . 

dr.       °  '  '' 

1  =  360  =  21600  =  1296000 


Oir. 


Scale— m,  60,  360. 


MlfeUKlJ^A.NKtJlj.^.  185 

1.  A   Qiiftdrdut  is  \  of  a  circumference,  or  90°;    a  Sr.rtdiif 
is  i  of  a  circumference,  or  60°. 

2.  The  length  of  a  degree  of  longitude  on  the  earth's  surface  at 
the  Equator  is  G9.1G  miles. 

:j.  In  astronomical  calculations  30°  are  called  a  Sign,  and  there 
are  therefore  12  signs  in  a  circle. 

EXERCISES. 

1.  How  many  minutes  are  there  in  5°?     6°? 

2.  In  35  degrees  how  many  seconds  are  there?     In  27°? 
Li  21°  12'  18"? 

3.  How  many  seconds  are  there  in  34°  12'  43"? 

4.  In  468560  seconds  how  many  minutes,  etc.,  are  there? 

5.  In  384500  seconds  how  many  minutes,  etc.,  are  there? 

6.  How  many  seconds  are  there  in  J  Cir.  ?     In  \1     In  ^? 

7.  How  many  minutes  are  there  in  2  quadrants?     In  2 
sextants? 

COUNTING. 

281.  The  following   denominations   arc    iisod   in   counting 
some  classes  of  articles: 

12  Things  =  1  Dozen    .    .    .    doz. 
12  Dozen  =  1  Gross      .    .    .    gr. 
12  Gross    =  1  Great  Gross    .    G.  gr. 

Two  tilings  arc  often  called  a  fxitr,  six  things  a  set,  and  twenty  things 
a  score;  as  a  pair  of  birds,  a  sei  of  spoons,  a  score  of  years. 


STATIONERS'  TABLE. 

2S2.  The  denominations  used  in  the  paper  trade  are: 

24  Sheets      =i  1  Quire. 

20  Quires     =  1  Ream. 

2  Reams     —  1  Bundle. 

r>   H.,n.llo«  --  1  Rnlo. 


186  DENOMINATE   NUMBERS. 

The  terms /ofoo,  quartOj  octavo,  applied  to  books,  indicate  the  number 
of  leaves  into  which  a  sheet  of  paper  is  folded.  Thus,  when  a  sheet 
of  paper  is  folded  into  2,  4,  8,  12,  16, 18,  or  24  leaves,  the  forms  are 
called  respectively,  folio,  4to,  or  quarto,  8vo,  or  octavo,  12mo,  16mo, 
18mo,  and  24mo. 

E  X  j^  jn  1  :i  m:  s. 

1.  How  many  eggs  are  there  in  5  dozen  ?   7  doz.  ?   10  doz.  ? 

2.  How  many  crayons  are  there  in  2  gross  ?    3  gr.  5  doz.  ? 

3.  How  many  things  are  there  in  a  great  gross? 

4.  What  will  be  the  cost  of  3  dozen  brushes  at  8.45  each  ? 
6.  A  man  lived  3  score  and  10  years.     What  was  his  age  ? 
6.  What  will  3  reams  of  paper  sell  for  at  $.15  per  quire  ? 


REDUCTION  OP  DENOMINATE  FRACTIONS. 

283.  The  principles,  processes  and  analyses  are  essentially 
the  same  as  those  of  denominate  integers. 

CASE    I. 

284.  To  rednee  cleiioniinate  fractions  to  equivalent 
nuiuberi$  of  lower  denominations. 

JBX  ERCISES. 

1.  How  many  hours  are  there  in  1  day?     In  ^  day?     In 
J  of  a  day?     In  J  of  a  day?    In  f  of  a  day? 

2.  How  many  ounces  are  there  in  ^  pound  avoirdupois? 
In^lb.?     Inilb.? 

3.  How  many  pints  are  there  in  ^  of  a  peck  ?     In  f  pk.  ? 

4.  How  many  pecks  and  quarts  are  there  in  f  of  a  bushel? 

5.  How  many  pounds  and  ounces  are  there  in  f  cwt.  ? 

6.  How  many  inches  are  there  in  |  of  a  foot  ?   f  ft.  ?   f  ft.  ? 


REDLCriO.N    UK   DKNOMINATK   FRACTIONS.  187 

7.  Change  ^  of  a  rd.  to  units  of  lower  denominations. 

PROCESS.  Analysis. — Since  in  1 

r,   .A'xxvd  —  U  vd  —  344vd       ^^  ^^^^^^  ^^^  ^^  y*'"''"' 
,  ot  Y  yd.  -  fi  ya.  _  6^  ya.      .^     ^^  ^  ^^  ^^^^^  ^^j,, 

1}  of  3    ft.  =  H  ft-    =  2H  ft.         be  ^  of  51  yards,  or  3^ 

U-  of  12  in.  =  i^  in.  =  9y\  in.       yards. 

Since  in  1  yard  there 
arc  3  feet,  in  ||  of  a  yard  there  will  be  j J  of  3  ft.,  or  2} J  ft. 

Since  in  1  foot  tlicre  are  12  inches,  in  {\  of  a  foot  there  will  be  {^ 
of  12  inches,  or  9i\  in. 

Therefore  ^  of  a  rod  is  equal  to  3  yd.  2  ft.  9^^  in. 

Change  the  following  to  lower  denominations: 


8.  f  of  a  pound  Troy. 

9.  4  <^^  *^  ^^' 

10.  i  ^^^  fiu-long. 

11.  -j^of  an  acre. 


12.  f  of  a  peck. 

13.  -^  of  a  day. 

14.  f  of  a  sq.  rd. 

15.  T^  of  a  cu.  yd. 


16.  Express  yj-j  of  a  gallon  as  a  fraction  of  a  gill. 

Analysis. — Since  in  1  gallon  there  are  32  gills,  in  y}^  of  a  gallon 
there  arc  j\^  of  32  gi.,  or  ^^j  gi.    Hence  j\^  gal.  =  /^^  of  a  gill. 

17.  Express  -^  of  a  bushel  as  a  fraction  of  a  pint. 

18.  Express  tAtf  °^  ^  ™*^®  ^  ^  fraction  of  a  fbot. 

19.  Express  -j-A^  of  a  pound  as  a  fraction  of  a  scruple. 

20.  Express  .006  of  a  bushel  as  a  decimal  of  a  pint. 

21.  Express  in  lower  denominations  .685  of  a  pound  Troy. 

PR0CE8S.  Analysis. — Since  in  1    pound  there  are  12 

g  g  5  ounces,  in  .685  of  a  pound  there  are  .685  of  12 

-  o  ounces,  or  8.220  ounces. 


Since  there  ai*e  20  pennyweights  in  1  ounce, 

8 .  2  2  0  OZ.  in  .220  of  an  ounce  there  are  .220  of  20  penny^ 

2  0  weights,  or  4.400  iM?nny  weights. 

iTJoO  pwt.  Since  in  1  jK^nny weight  there  are  24  grains, 

*     n4  '"  -4^  <*f  *  l^ennyweight  there  are  .400  of  24 

grains,  or  9.600  grains. 

9.600  gr.  Tlxn-t'or..  (;k.mi.  i-.......,i  ,,,«...    »  j,v.  .n:._,r. 


188  DENOMINATE   NUMBERS. 

Express  in  units  of  lower  denominations: 


22.  £.575. 

23.  .1935  of  a  pound  Troy. 

24.  .436    of  a  ream. 

25.  .1845ofagalJon. 


26.  .135   of  a  rod. 

27.  .455   of  a  mile. 

28.  .4832  of  a  bushel. 

29.  .684   of  a  league. 


CASE  II. 

2S5.  To  clinnji^e  clciioiiiiiinto  fractions  to  eqiiivaleut 
fl*acUous»  of  higher  deiioiiiiiiali<»ii». 


1SXERCI8E8. 

1.  What  part  of  a  pound  Troy  is  1  ounce?    Is  f  oz.?    Is 
i  oz.? 

2.  What  part  of  a  ton  is  1  pound  ?    Is  ^  lb.  ?    Is  ^  lb.  ? 

3.  What  part  of  a  mile  is  1  rod  ?    Is  J  rd.  ?    Is  |  rd.  ? 

4.  What  part  of  a  league  is  1  mile  ?    Is  1  rd.  ?    Is  ^  rd.  ? 

5.  What  part  of  an  hour  is  J  of  a  minute  ?   Is  |^  of  a  min.  ? 

6.  What  part  of  a  week  is  ^  of  a  day  ?    f  of  a  day  ? 

7.  What  jmrt  of  a  bushel  is  f  of  a  pint? 

PROCESS.  Analysis. — Since  there  are  64  pints  in 

a  bushel,  1  pint  is  ^^  of  a  bushel,  and  ^ 
of  a  pint  ia  f  of  ^y  of  a  bushel,  or  ^|y 
of  a  bushel.    Or, 

Since  we  are  required  to  change  pints  to 
bushels  we  have  an  example  in  reduction 
ascending,  and  hence  we  divide  by  2,  8,  and 
4,  respectively. 

8.  Reduce  ^  of  an  inch  to  the  fraction  of  a  yard. 

9.  Change  f  of  a  second  to  the  fraction  of  an  hour. 

10.  Express  .375  of  a  week  as  a  fraction  of  a  year. 

11.  Express  .35  of  a  pound  as  a  fraction  of  a  ton. 

12.  Express  f  of  a  cubic  inch  as  a  fraction  of  a  cubic  foot. 


*x,^= 

rfrbu. 

Or 

4^ 

-  2  = 
-8  = 
-4  = 

Aqt- 

REULC11U>    OK    J»J:M).M1NA11:    FltACTlONS.  189 

13.  Change  ^  of  a  stjuare  yard  to  a  fmction  of  an  acre. 

14.  Reduce  f  of  a  pint  to  a  fraction  of  a  barrel. 

a\SE  III. 

286.  To  express  one  deuominate  number  as  a  iruc- 
tioM  <>r»iiotlier. 

1.  What  part  of  a  foot  are  3  in.  ?    6  in.  ?    9  in.  ? 

2.  What  part  of  an  hour  are  30  min.?     15  min.  ?     45 


nun. .' 


3.  What  part  of  a  gallon  is  1  pint?     2  pints?    1  quart? 

4.  AVhat  part  of  a  gallon  are  2  quarts?     2  qt.   1  pt. ? 
3qt.  1  pt.? 

5.  ,What  part  of  3  ft.  6  in.  are  2  ft.  3  in.  ? 

Analysis— Since  3  ft.  6  in.  =  42  in.,  and  2  ft.  3  in.  =  27  in.,  27  in. 
=  J5of  42in. 

6.  What  part  of  3  yd.  2  ft.  are  2  yd.  2  ft.  ? 

7.  What  part  of  5  gal.  3  qt.  1  pt.  are  2  gal.  1  qt.  1  pt.  ? 

8.  What  jxirt  of  2  pounds  Troy  are  3  oz.  10  pwt.  ? 

9.  What  part  of  3  pecks  are  2  qt.  1  pt.  ? 

10.  What  part  of  3  barrels  are  13  gal.  3  qt.  2  pt.  2  gi.? 

11.  Express  15s.  7d.  in  the  decimal  of  a  ix)und  sterling: 

1st.  process.  Analysis. — In  order  to  find  what  part 

-  e     »j  -I  OT  J         one  number  is  of  another,  both  must  be 

108. 7a. =     187a.         j      j  .    *.  j         •    .•        i- 

reduced  to  the  same  denomination,     las. 

£1          =      240d.  7d.  =  187d.    and    XI  =  240d.    Therefore 

£||^      =£.7791-1-  187d.  =  £.Hi,  which,  reduced  to  a  deci- 
mal, is  equal  to  £.7791  -\- .    Or, 

-I).  PROCESS.  Analysis. — Since  7d.  is  ^^^  of  a  shil- 

1  9^7/^  ^*"S>  i'  ™*y  ^  reduced  to  a  decimal  by 

annexing  ciphers  to  the  numerator  anil 

.6833  +  8.  dividing   by   12,  which  gives  .6833  +  8. 

9ftM»;    r;ftQQ_.c  Therefore  15s.  7d.  =  15.5833  +  8. 

Zy))liy.0l566-\-8.  gj^^    J   BhiWmis   is   A   of   a  pound, 

£.7791-}-  1 5.r,8:r>  -i  h.  -:-  £^^-^1^,  or  £  .7791  -^ . 


190  DENOMINATE    NUMBERS. 

12.  Reduce  4  hr.  15  min.  to  the  decimal  of  a  day. 

13.  Reduce  3  pk.  2  qt.  to  the  decimal  of  a  bushel. 

14.  Reduce  3  ft  6  in.  to  the  decimal  of  a  rod. 

15.  Reduce  188.  5Jd.  to  the  fraction  of  a  pound. 

16.  Reduce  18s.  5}d.  to  the  decimal  of  a  pound. 

17.  Reduce  16  lb.  11  oz.  to  the  fraction  of  a  hundred-weight. 

18.  Reduce  37  rd.  14ft.  3  in.  to  the  decimal  of  a  mile. 

19.  Reduce  3  da.  5  hr.  14  min.  to  the  decimal  of  a  week. 

20.  Reduce  8  quires,  15  sheets,  to  the  decimal  of  a  ream. 

21.  Change  3  cd.  ft.  7  cu.  ft.  to  the  decimal  of  a  cord. 

22.  Change  654  yd.  9  in.  to  the  decimal  of  a  mile. 

23.  Chang!    '         "  i.wt.  1'^  gr.  to  the  fraction  of  a  pound 
Troy. 

.  24.  Write  rules  Ibr'^each  of  the  cases  in  denominate  num- 
bers. 

REVIEW  EXERCISES. 

287.    1.  What  will  be  the  cost  of  15  lb.  8  oz.  of  butter 
at  ?.31  per  pound? 

2.  What  must  Ix'  i)aicl  lur  o  pk.  2  qt.  of  IxTries  at  9  cents 
a  quart? 

3.  Mr.  A.  sold  18  bu.  3  pk.  of  barley  at  $1.05  per  bushel. 
How  much  did  he  get  for  it? 

4.  How  much  must  be  paid  for  making  42  rd.  7  ft.  8  in. 
of  fence  at  S  .75  per  foot? 

5.  How  much  butter  at  $  .30  a  pound  must  be  given  for 
12  gal.  3  qt.  of  molasses,  at  8  .50  per  gallon? 

6.  Bought  15  bu.  of  oats  at  8  .37^  a  bushel,  and  sold  them 
at  15  cents  a  half-peck.     How  much  did  I  gain? 

7.  How  many  cords  of  wood  are  there  in  a  pile  4  ft.  wide, 
6  ft.  high,  60  ft.  long?     What  would  it  cost  at  $4.25  a  cord? 

8.  A  man  built  a  cistern  10  ft.  long  and  6  ft.  wide,  that 
wouimiold  100  barrels.     How  hisrh  did  he  make  it? 


\hUi   l.^l> 


101 


9.   W.hal  is  gaiiK'd  l)y  selling  1  oz.  Troy  ol"  opium  i'ur  SI, 
which  was  purchased  at  the  rate  of  $.75  per  oz.  Avoirdupois? 

10.  What  are  the  contents  of  a  field  15  rd.  8  ft.  wide,  27 
rd.  9  ft.  long?    What  is  its  value  at  §150  per  acre? 

11.  How  many  days  of  10  hours  each  will  it  require  to 
make  a  million  marks  if  I  make  2  per  second? 

12.  Wiiat  is  the  value  of  a  plank  18  ft.  long,  16  in.  wide, 
and  4  in.  thick,  at  $18  per  M? 

13.  If  at  10  cents  a  foot  the  Atlantic  cable  cost  $1689600, 
what  is  its  length? 

14.  A  druggist  put  up  7^  83  49  in  two-grain  pills.  How 
many  pills  did  he  put  up? 

15.  Bought  paper  at  $2.55  per  ream  and  sold  it  at  20 
cents  i)er  quire.     How  much  did  I  gain? 

16*-  How  much  sugar  at  12  cents  a  pound  can  be  obtained 
for  13  lb.  7  oz.  butter  at  27^  cents  a  pound? 

17.  A  farmer  sold  3  piles  of  wood  at  $4.60  per  cord.  The 
following  are  the  dimensions  of  the  piles:  The  first  was  73 
ft.  9  in.  long,  6  ft.  high,  and  4  ft.  wide;  the  second  was  30 
ft.  long,  7  ft.  2  in.  high,  and  4  ft.  wide;  the  third  was  37  ft. 
long,  3  ft.  6  in.  high,  and  4  ft.  wide.  How  much  should  he 
receive  for  his  wood? 

18.  A  printer  used  4  reams  8  quires  12  sheets  of  paper  for 
half-sheet  posters.  How  many  did  he  print?  What  did 
they  cost  at  $6.50  \)er  M? 

19.  Hay  at  $18  |)cr  ton  is  exchanged  for  flour  at  $6.85  per 
barrel.     How  many  barrels  are  txjual  to  a  ton? 

20.  Two  men  who  are  equal  jwirtners,  obtained  from  a  field 
327  bu.  3  j)k.  5  qt.  of  oats.  One  of  them  claimed  167  bu. 
3  pk.  for  his  share.  Did  he  claim  too  much  or  too  little? 
H<nv  much? 

J 1 .  A  cubic  foot  of  water  weighs  about  62  lb.  8  oz.  What 
will  be  the  weight  or  pressure  on  a  sfpiaro  yanl  whore  the 
sea  is  20  fathoms  deep? 


192  DENOMINATE   NUMBERS. 


ADDITION. 

288.  The  processes  of  adding,  Miuiia*  img,  multiplying, 
and  dividing  compound  numbers  are  based  ujwn  the  same 
principles  as  those  governing  similar  operations  in  simple 
numbers. 

The  only  difference  between  the  processes  Ls  caused  by  com- 
fK>nnd  numlx^rs  having  a  varying  ecale^  while  simple  numbers 

BXEJtCISES. 

1.  What  is  the  sum  of  130  rd.  5  \  1  1  li.  (l  in.,  215  rd. 
2  ft.  8  in.,  304  rd.  4  yd.  11  in.? 

PROCESS.  Analysis. — The  numbers  should 

be  written  as  in  simple  addition,  so 
tliat  units  of  the  same  denomina- 
tion stand  in  the  same  column,  and 
for  convenience  we  begin  at  the 
right  to  add. 

The  sum  of  the  inches  is  25  in., 
which  is  equal  to  2  ft.  1  in.     We 
write  the  1  under  the  inches  and 
2  mi.       10        5        0         7        'uld  the  2  ft.  with  the  feet.     The 

sum  of  the  feet  is  5  ft.,  or  1  yd.  2  ft. 
AVe  write  the  2  as  feet  in  the  sum  and  add  the  1  yd.  with  the  yards. 

The  sum  of  the  yards  is  10  yd.,  or  1  rd.  A\  yd.  We  write  the  4^  yd. 
as  yards  of  the  sum,  and  add  the  1  rd.  with  the  rods.  The  sum  of  tlic 
rods  is  650  rd.,  or  2  mi.  10  rd.,  which  we  write  as  miles  and  rods  of  the 
sum. 

Therefore  the  sum  is  2  mi.  10  rd.  4^  yd.  2  ft.  1  in.  Or,  since  \  yd. 
equals  1  ft.  6  in.,  the  sum  may  be  expressed  as  2  mi.  10  rd.  5  yd.  7  in. 

Rule. — Chaise  the  rule  for  i)ie  addition  of  simple  numbers 
80  that  it  may  he  applicable  to  denominate  numbers. 


nl. 

vd.         h. 

in. 

130 

1 

li 

215 

0       2 

8 

304 

4       0 

11 

2  mi.      10 

H    2 

1 

Or, 

*=1 

6 

ADDIIION.  193 

2.  What  is  tlu3  sum  of  12  11).  i)  uz.  l.i  pwt.,  21  11).  <S  oz. 
15  pvvt.,  13  lb.  7oz.  10  pwt,  51  lb.  3  oz.  17  pwt.? 

3.  What  is  the  sum  of  £71  6s.  ojd,  £32  8s.  5|d.,  £61 
15s.  8Jd.,  £37  18s.  SJd.,  £115  lis.  7d.? 

4.  Find  the  sum  of  10  mi.  217  rd.  2  yd.  3  ft.  4  in.,  7  mi. 
185  rd.  3  yd.  9  in.,  19  mi.  37  rd.  6  yd. 

5.  Find  the  sum  of  3  T.  7  cwt.  39  lb.  8  oz.,  8  T.  11  cwt. 
48  lb.,  11  oz.,  13  T.  33  lb.  10  oz.,  9  cwt.  18  lb.  9  oz. 

6.  Find  the  sum  of  18  gal.  3  qt.  1  pt.  3  gi.,  15  gal.  2  qt. 

1  pt.  2  gi.,  11  gal.  2  qt.  2  gi.,  3  qt.  1  pt.  1  gi. 

7.  A  miller  bought  four  loads  of  grain  containing,  respect- 
ively, 25  bu.  3  pk.,  28  bu.  2  pk.,  32  bu.  3  pk.  5  qt.,  2S  bn. 

2  pk.  7  qt.     How  much  grain  did  he  buy? 

8.  How  much  wood  is  there  in  3  piles  containing,  respect- 
ively, 37  C.  21  cu.  ft.  1140  cu.  in.,  29  C.  110  cu.  ft.  708 
cu.  in.,  and  34  C.  121  cu.  ft.  398  cu.  in,? 

9.  Find  the  sum  of  -  mi.,  .35  rd.  and  22  rd. 


Analysi.s. —  Each  of  the 
fractions  Is  expressed  in  in- 
tegers of  lower  denominations, 
and  then  they  are  added. 


rufx:Es.>. 

nl. 

ft. 

in. 

f  mi.  —  l.)7 

•  ) 

4? 

.35  rd.  -^ 

. ) 

i'  i^;- 

21  rd.  =       2 

6 

2i 

130     11       3fH 


10.  A  merchant  .sold  12J  yards  of  cloth  to  one  person,  8J 
yards  to  another,  37.V  yards  to  another,  39J  yards  to  another. 
How  many  yards,  feet  and  inches  did  he  sell  ? 

11.  What  is  the  amount  of  land  in  the  following  lots,  the 
first  containing  ^  of  an  acre,  the  second  J  of  an  acre,  the 
third  129J  sq.  rd.,  and  the  fourth  118^  .s^i.  rd.? 

12.  A  merchant  sold  the  following  quantities  of  molasses, 
viz:  On  June  15,  24  gal.  2  qt.  3  pt. ;  Jime  16,  45|  gal.; 
June  17,  }\  1.1.1.  i:]9l  ga^O  ^low  much  did  he  sell  in  that 
time  ? 

13 


r.M  DKXOMIN ATE   NUMBERS. 

13.  James  i  18  da.  old,  Henry  is  2  yr.  ^  m.  . 
6  (la.  older  tluin  Jjiiiu *s,  William  is  7  yr.  10  mo.  24  da.  older 

than  Henry,  iw^  U.  . •].... t  ;.  on l.L.r  thnn  Williai!'     TT-- 

old  is  Herbert 

14.  Find  the  Mini  oi'  2(^  ewL,  IGf  T.,  17^  lb.,  li>  cwL 
18  lb.  7  oz.,  15  lb.  8  oz.,  2  T.  7  lb.  5  oz.,  f  lb.,  J  T.,  2  T. 
3  cwt  57  lb.  4  oz. 


SUBTRACTION. 

d.  J  yd.  1  iL  7  ill.,  subtract  100  rd. 

Analysir. — The  numbers  nhouUl  be 
^mtten  an  in  Kiniplc  Hubtraction,  bo 
that  units  of  tlie  Aaiue  onlor  stand  in 
the  same  cohiran,  and,  for  convenience, 
Ix?gin  at  tl»e  right  to  subtract. 

Since  we  can  not  subtract  9  in.  from 

7  in.,  we  unite  with  7  in.  a  unit  of  the 

next  higher  order,  making  1  ft  7  in., 

26        4       0       4         or  19  in.    Then  9  in.  from  19  in.  leaves 

10  in.,  which  we  write  as  inches  in  the 

remainder.     Inasmuch  as  1  ft.  was  united  with  7  in.,  there  arc  no  feet 

remaining  in  the  minuend. 

Since  we  can  not  subtract  2  ft.  from  0  ft,  we  unite  with  0  ft  a  unit 
of  the  next  higher  order,  making  3  ft.  Then  2  ft  from  3  ft  leaves  1  ft., 
which  we  write  as  the  feet  of  the  remainder. 

Since  4  yd.  can  not  be  subtracted  from  2  yd.,  we  unite  witli  2  yd.  a 
unit  of  the  next  liigher  order  and  proceed  as  before.  The  remainder 
wiU  be  2G  nl.  4  yd.  0  ft.  4  in. 

Rule. — Change  (he  rule  for  subtraction  of  iimple  numbera  so 
timt  it  viay  be  applicable  to  denominate  numbers. 

2.  From  2  mi.  116  nl.  4  yd.  0  ft.  4  in.,  take  1  mi.  120  rd. 
2  yd.  1  ft.  8  in. 


289.  1. 

,  Froni   l: 

.'7  1 

4  yd.  2  ft  9  in 

PROCESS. 

rd. 

yd.     ft. 

in. 

127 

3       1 

7 

100 

4      2 

.9 

26 

3V     1 

in 

Or. 

i=i 

t) 

SUBTRACTION.  195 

From  If)  cwt.  37  11).  10  oz.,  take  8  cwt.  42  lb.  8  oz. 
From  1  hlul.  38  gal.  3  qt.  2  pt.,  take  GO  gal.  2  qt.  1  gl 
From  13  lb.  8  oz.  13  pwt.  15  gr.,  take  8  lb.  8  oz.  16  pwt. 


0. 

From  18°  33'  16" 

,  take  9° 

42'  28". 

7. 

From 

37  C. 

7  cd 

.  ft.  11  cu.  ft.,  take  18  C. 

7cd. 

ft. 

V_>  <'u 

.  ft. 

From 

J  bbl. 

take 

7igal. 

J  bbl.  = 
7|gal.= 

PROCESS, 
gal.      qt. 

=  23      2 
:     7       2 

pt.      gi. 
1 
0      3i 

Analysis. — The    fractions 
are  first  expressed  in  integers 
of  lower  denominations  nnd 
then  subtracted. 

16      0      0        4 

9.  From  J  of  an  acre  of  land  a  piece  containing  72  sq.  rd. 
160  sq.  ft  39  sq.  in.  was  sold.     How  mucli  was  left? 

10.  A  mercbant  sold  clotb  for  £384  6s.  5^d.  which  cost 
him  £297  9s.  83d.     How  much  was  his  profit? 

11.  From  a  farm  of  285  acres  there  were  sold  at  one  time 
97J  acres,  and  at  another  38  A.  39J  sq.  rd.  How  much  was 
left? 

12.  A  merchant  l)ought  9  reams  18  quires  15  sheets  of 
pajx^r,  from  which  ho  sold  r>,-^,  reams.  How  nmoh  remained 
unsold? 

V).  How  long  was  it  from  Jan.  10,  1841,  to  May  7,  1853? 

PROCESS.  An.\ LYSIS. — Since  the  later  date  expresses 

1853        5  ~        ''"  ^''t'Jitfr  l^eriod  of  time,  we  write  it  as  the 

1  oi^        1        IJI        minuend,  and  the  earlier  date  as  the  subtra- 

hend,  pivinj;:  the  month  its  number  instead 

1  J        :;        I'T        of  tlie  name.     We  then  subtract  as  in  de- 
nomiiiato  numbers,  considering  30  days  one 
month,  and  12  months  one  year.    The  remainder  will  be  the  time  as 
forrect  as  it  can  Ix?  expressed  in  months  and  days. 

14.  How  long  was  it  from  Jan.  3.  1843.  to  Mar.  15,  1851? 


196  DKNOMINATE   NUMBERS. 

15.  How  old  was  a  man  who  was  1  - m  April  2,  1808,  and 
who  died  Dec.  15,  ISOir 

16.  A  man  bought  a  liirm  .May  15,  l^U,  and  pai<i   :   r  ir 
Jan.  5,  1871.     How  long  did  it  take  to  pay  for  it? 

1 7.  A  legacy  of  $3000  was  to  be  paid  to  a  man  3  yr 
5  da.  atier  Dec.  8,  1837.     When  was  it  to  be  paid? 

18.  How  many  years,  months  and  days  from  the  .l:i\  -i 
your  birth?  or,  How  old  are  you 

li).  The  American  Civil  War  inMrjui  Aj)nl   11,  l^•il.  and 
ended  April  9,  1865.     How  long  did  it  continue? 

20.  A  note  dated  July  9, 1871,  was  paid  October  KK  1   :• 
II  w  long  did  it  run  before  it  was  jiaid? 


MULTIPLICATION. 

290.    1.  How  much  is  5  times  147  rd.  4  yd.  2  ft.  8  in.? 

rROCEss.  Analysis. — We  write  the  numbers  as 

rl       vd    ft.    in.       '"  j^imple  numbers,  and  for  convenience 
1  .  -     "  I      9      ^        W'/xn  at  the  right  to  multiply. 

')  times  8  in.  are  40  in.,  or  3  ft.  4  in. 

We  write  the  4  in.  as  inches  in  the  prnd- 

2  mi.    99     2     1      4       "ct,  and  reserve  the  3  ft.  to  add  with  the 

product  of  feet. 
r»  times  2  ft  are  10  ft.;  10  ft.  -f  3  ft.  reserved  equal    13  ft.,  or  4  yd. 
1  ft.     We  write  the  1  ft.  in  the  prod  '  reserve  the  I  y<1.  (..  add 

to  the  product  of  yards. 

5  times  4  yd.  equal  20  yd.;  20  yd.  -r  4  yd.  reser%ed  equal  24  yd.,  or 
4  rd.  2  yd.  We  write  the  2  yd.  in  the  product  and  reserve  the  rods  to 
add  to  the  product  of  rods. 

5  times  147  rd.  are  735  rd.;  7., ,  ....  1  rd.  r-  -'  rv. ,]  ,  Mual  739  rd., 
or  2  mi.  99  rd.,  which  we  write  in  the  produt 

Therefore  the  product  is  2  mi.  99  rd.  2  yd.  1  it.  ^  m. 

Rule. — Modify  the  rule  for  multiplication  of  simple  numbers 
80  tJiat  it  may  be  applicahle  to  denominate  numberi<. 


DIVISION.  li)7 

2.  Multiply  *J  gal.  o  qt.  1  pt.  3  gi.  by  7. 

3.  Multiply  17  lb.  8  oz.  3  pwt.  15  gr.  by  9. 

4.  Multiply  1  T.  4  cwt.  35  lb.  6  oz.  by  10. 

5.  A  farm  consists  of  7  fields  each  containing  18  A.  25 
sq.  rd.     How  much  land  does  it  comprise  ? 

6.  What  is  the  length  of  a  fence  which  encloses  a  square 
field  each  side  of  which  is  28  rd.  5  yd.  2^  ft.  long? 

7.  How  much  wood  is  there  in  7  piles,  each  containing 
13  C.  7cd.  ft.  24  cu.  ft.? 

8.  What  will  14J  yd.  of  lace  cost  at  £2  5s.  6d.  per  yard? 

9.  What  is  the  value  of  4  loads  of  potatoes,  each  contain- 
ing 27  h»i.  8  pk.,  ;it  ?.4o  per  bushel? 


DIVISION. 

291.    1.  Divide  27  bu.  3  pk.  5  qt.  1  pt.  into  6  equal  parts. 

PROCESS.  Analysis. —  Since  the  quan- 

o;v7  bu.  3pk.   5qt.    1    pt.        tit  vis  to  be  divided  into  6  equal 

" parts,   each   part  will    contain 

*  ^  ^  ^tT  o?i«-»jj:</i  of  the  quantity. 

One-sixth  of  27  bu.  is  4  bu. 
witli  a  reiuaiiuUr  »>i  ;)  l>ij.  We  write  the  4  bti.  in  the  quotient  and 
unite  the  3  bu.  remaining  with  the  number  of  the  next  lowest  denomi- 
nation, making  15  pk. 

One-sixth  of  15  pk.  is  2  pk.  and  3  pk.  remaining.  We  write  the 
2  pk.  in  the  quotient,  and  unite  the  3  pk.  remaining  with  the  number 
of  the  next  lower  denomination,  making  29  qt. 

One-sixth  of  29  qt.  is  4  qt.  and  5  qt.  remaining.  We  write  the 
4  qt.  in  the  quotient,  and  unite  the  5  qt.  witli  the  number  of  the  next 
lowest  denomination,  making  11  pt. 

One-sixth  of  11  pt.  is  1^  pt.,  which  we  write  in  the  quotient 
Therefore  the  quotient  is  4  bu.  2  pk.  4  qt.  1§  pt. 

RuiJC. — Change  ihe  ndc  for  Ow  division  of  simple  nwnbei'S  80 

unit   \t   )H(lft  hr  (iitulirnhlr  fa  driiniinutifr   »»//))>/)//•,•<. 


198  DENOMINATE    NUMBEIiS. 

2.  In  8  bags  th<  i  ,  lui.  :;  j.k.  \  .jt.  How  much  docs 
each  bag  conUiin? 

3.  A  -.  '  :  .livi.ltMl  hi.s  farm  of  4->7  A.  131  sq.  rd. 
equally  an                 ">  sons.  What  was  the  share  of  each  ? 

I.   A  brewer  tilhd  4  ca-ks  of  icjual  sire  from  a  \ 
taining  315  gaL  3  qt.     How  larL'e  was  each  cask? 

5.  16  T.  1300  lb.  of  hay  was  drawn  at  9  loads.  Wliat 
was  the  average  weight  per  load? 

G.  If  a  pile  of  wood  containing  8  cords  100  cu.  ft.  be 
equally  dividetl  among  3  persons,  how  much  will  each  receive? 

7.  When  £31  58.  8d.  is  divided  equally  among  10  jiersons, 
how  much  does  each  receive  ? 

8.  If  31  cwt.  18  lb.  of  tea  is  put  up  in  packages,  each 
containing  3  lb.  8  oz.,  how  many  packages  will  there  be? 

PROCGBS.  Analysis.— Since 

.^1  rwt.  18  1b.  =4988  8  oz.      *''^  *^'''^'^'"  ^"^  ^*^^ 

dividend    arc   Muiilar 

^  Ih.  8  0Z.=  56    oz.       denominate    numbers, 

4988  <^  i;  oz.  =        8 9 Of  we  may  reduce  them 

to  the  8amc  denomi- 
nation, and  then  proceed  to  divide  a^  in  Kiraple  numberR. 

9.  How  many  times  must  a  man  dip  with  a  dipper  hold- 
ing 2  iji.  1  { >r.  >'  ilia  I  i  cask  containing 
31  gal.y 

10.  If  a  man  walks  at  an  average  rate  of  23  mi.  160  rd.  4* yd. 
2  ft.  j)er  day,  how  long  will  it  take  him  to  walk  100  miles? 

11.  If  a  man  can  travel  300  mi.  in  1-  '  ■  how  far  can 
he  travel  daily? 

12.  How  many  barrels  of  sugar,  each  containing  2  cwt. 
35  lb.,  are  there  in  3  T.  4  cwt.  18  lb.? 

13.  How  many  spoons,  each  weighing  2  oz.  10  pwt.,  can 
be  made  from  13  lb.  7  oz.  15  pwt.  of  silver? 

14.  How  many  pickets  2  ft.  4  in.  long  and  2  in.  wide,  can 
be  made  out  of  5  boards  each  1 1  ft.  8  in.  long  and  8  in.  w  ide  ? 


IX)NCJ1TU1)E   ASV   TIME.  199 


LONGITUDE  AND  TIME. 

292.  1.  Where  does  the  sun  appear  to  rise? 

2.  How  long  will  it  be  before  it  rises  again? 

3.  Through  how  many  degrees  of  space  does  it  appear  to 
pass  in  this  daily  motion?  Am.  360°. 

4.  Since  it  seems  to  travel  360°  in  one  day,  or  24  hours, 
how  great  will  be  its  apparent  motion  in  1  hour?  ' 

5.  If  the  earth  moves  15°  in  1  hour,  how  far  will  it  move 
in  1  minute? 

6.  If  it  moves  15'  in  one  minute  of  time,  how  far  will  it 
move  in  1  second? 

7.  How  does  the  number  of  degrees  passed  over  compare 
with  the  number  of  hours?  The  number  of  minutes  of  space 
with  the  number  of  minutes  of  time?  The  number  of  seconds 
of  space  with  the  number  of  seconds  of  time? 

8.  When  it  is  sunrise  at  New  York,  how  long  will  it  l)e 
before  it  is  sunrise  at  a  place  15°  west  of  New  York?  30° 
west?    45°  west?    60°  west? 

9.  When  it  is  sunrise  at  New  York,  how  long  l)efore  was 
it  sunrise  at  a  place  15°  east?     30°  east?     45°  east? 

10.  When  it  is  sunrise  at  any  place,  how  long  will  it  be 
before  it  is  sunrise  at  a  plnrc  15°  west?  15°  east?  30°  west? 
30°  cast? 

11.  When  it  is  noon  at  any  place,  what  time  is  it  at  a  place 
15°  west?     15°  east?     30°  west?     30°  east? 

12.  If  I  travel  eastward  will  my  \vatch  become  too  slow  or 
Uh)  fast?     If  I  tnivel  westward  what  change  will  take  place? 

13.  What  places  have  sunri.se  at  the  same  time?  Noon 
at  the  same  time?     Midnight  at  the  same  time? 

293.  A  Merhliaii  is  an  inuiginary  line  passing  from  the 

V'«r!!-  Pole  to  the  Smith  Polo  thronirh  ;iiiy  place. 


200  DENOMi.NA  1  i;    M  Mi;i:i;-. 

294.   Louffitude  is  the  diiJtaiKr  (a>i  -  i  ,  I'rora  a 

given  meridian. 

RELATION   BETWEEN   LONGITUDi:    \M»    ilME. 

•iiritiidc  make  1  H<»ur  (liiUrvinv  in  time. 
15^  make   1  Minute  diflerence  in  time. 

\b^'  make    1  Second    difference  in  time. 

1®  nmkes  4  Minutes  difference  in  lime. 

I''  gi  makes  4  Second*  difference  in  time. 

117.  N      r.XEHCISF.S. 

1.  T  lu'  i«>ii'_:iiiMif  i»i    ifip>i<iii  1^   I  1     »j    o')     ^»\.-i,    111. II  "i   t..  ill' 

ciimati,  84°  29'  31"  west.     Wliat  is  the  diflerence  in  time? 

PRocf>w.  Analysis. — We  first  find  the  differ- 

„  ence  in  longitude  of  the  two  places,  and 

o4        jy       ol  since  there  are  15  times  a,s  many  de- 

7  1                    •>  0_  grecB,  minutes  and  seconds  as  there  are 

1  5  J 1  .*?         _             1  hours,  minutes  and  seconds  of  time,  wo 

,^3       44*  ®"*'  ^  ^^  ^^  ^'  ^'''  ^^^^^  '**  ^  "**"• 

2.  \\  iit'ii  it  t?  12  o'clock  M.  ill  1  M.i:i(ui|»iiia  it  is  •)  o'clofk, 
10  min.  P.  M.  at  Paris.  What  is  the  longitude  of  Paris, 
the  longitude  of  Philadelphia  being  75°  10'  west? 

PROCESS.  Analysis. —  Since  in   1 

5  hr.  1  0    min.  hour  the  earth  moves  15^ 

15  of  distance,  in  1  minute  lo' 

^-;r^      oTw  j-/r  •     T  of  distance,  in  1  second  15'^ 

/  /        30   difference  m  Long.         ^^  j,^^^^^^  ^^^  m^v^nc^ 

/  '^  -^  0  in    longitude    will    be    15 

2  ®      20'  east,  Long,  of  Paris.        t»nif«  as  many  degrees,  min- 
utes and  seconds  of  distance 
as  there  are  hours,  minutes  and  seconds  of  time.    Since  Philadolpliia 
is  75°  lO'  west,  and  the  difference  is  77°  W^  the  longitude  of  Paris 
is  2^  2(y  east. 


LOMJITl DK    AM)    TIME.  20l 

To  fuui  the  difference  in  time  when  the  difference  in  longi- 
tude is  given : 

Divide  Uie  difference  in  loncpiudey  expressed  in  degrees,  etc. ,  bij 
15;  tlie  several  quotients  will  be  tlie  difference  in  time  in  hours, 

ill i)iiif )'.•<,  (1)1(1  !<rrond.^. 

T<j  liiid  die  (lilicit'iRv  ill  loiigitiulo  wlien  tlio  differoncc  in 
time  is  given : 

Multiply  the  difference  in  time,  expressed  in  hours,  mimttes  and 
seconds,  by  Id;  the  several  products  will  he  tlie  difference  in  longi- 
tude, in  degrees,  minutes,  and  seconds. 

.3.  Two  places  are  32°  18'  24"  apart.  What  is  the  differ- 
ence in  time  between  them? 

4.  When  it  is  noon  at  San  Francisco  it  is  3  hr.  U  inin. 
7  sec.  P.  M.  at  Philadelphia.  What  is"  the  longitude  of  San 
Francisco  if  that  of  Philadelphia  is  75°  10'  west? 

5.  New  York  is  74°  3'  west  longitude,  and  Paris,  France, 
is  2°  2(/  east.  How  much  earlier  is  it  sunrise  in  Paris  than 
in  New  York? 

G.  Washington  is  77°  west  of  Greenwich,  EuLdand.  What 
is  their  difference  in  time? 

7.  When  it  is  noon  at  Washington,  wliich  is  77^  west, 
what  time  is  it  at  New  York,  which  is  74°  3'  west? 

8.  The  difference  in  time  l)etween  Halifax,  Nova  Scotia, 
and  Charleston,  S.  C\,  is  1  hr.  5  min.  ^  sec  What  is  their 
difference  in  longitude? 

9.  Pekin,  China,  is  110°  27'  r>0"  east  longitude,  and  Wasli- 
ington  is  77°  west  longitude.  When  it  is  noon  on  January 
Ist  at  Washington,  what  time  is  it  at  Pekin? 

10.  A  gentleman  traveling  found,  on  arriving  at  his  des- 
tination, that  his  watch,  which  kept  correct  time,  was  1  hr. 
11  min.  slow.  Which  way  was  he  traveling?  How  far  had 
ho  tnivdod  ? 


METRIC  SYSTEM 


THiiimimiiMiHii, 


ympif*^ 


Hiuittm 


295.  The  Metric  St/sfem  of  weights  and  measures 
has  been  legalist  by  the  United  States,  most  of  the  coun- 
tries of  Europe,  and  several  countries  of  Central  and  South 
America, 

Although  this  system  is  extremely  valuable  on  account  of  its  sim- 
pHcity,  it  is  not  in  general  use  in  this  country,  and  hence  is  not  treated 
as  fully  here  as  the  other  divbions  of  Denominate  Numbers. 

2%.  The  Unit  of  Lengthy  called  the  Metre  {meeter), 
from  which  the  system  derives  its  name,  is  nearly  one  ten- 
millionth  of  a  quadrant  of  the  earth's  circumference. 

297.  The  Unit  of  Area,  called  the  Are  (air),  is  a 
square  whose  side  is  10  metres.  It  contains  100  square 
metres. 

298.  The  Unit  of  Solidity,  caUed  the  Stere  (stair), 
is  a  cube  whose  edge  is  one  metre. 

299.  The  Unit  ofCa2)aciti/,  called  the  TJtre  (leeter), 
contains  a  volume  equal  to  that  of  a  cube  whose  edge  is  one- 
tenth  of  a  metre. 

300.  The  Unit  of  Weight,  called  the  Gramme,  is 

the  weight  of  a  cube  of  distilled  water  whose  edge  is  one- 
hundredth  of  a  metre. 

It  must  be  weighed  in  a  vacuum  and  at  the  period  of  its  greatest 
density,  39.2  Fahrenheit. 
(202) 


METRIC    SYSTEM.  203 

301.  From  these  standard  units  are  derived  the  multiples 
and  sub-niuhiplcs  which  are  named  to  express  units  of  higher 
or  lower  orders  in  the  decimal  scale.     Thus, 

For  multiples,  Greek  numerals  are  used: 

Deka,  10;   Hecto,  100;    KUo,  1000;   Myria,  10000. 

For  sub-multiples  the  Latin  ordinals  are  used: 
Deci,  10th;   Centi,  100th;   Milli,  1000th. 


Dekametre  . 
Dekagram  me 
Hectometre  . 
Kilolitre.  . 
Myriagramme 
Centigramme 
MilllErramme 


means  10  Metres. 

"  10  Grammes. 

"  100  ISIctres. 

1000  Litres. 

"  10000  Grammes. 

"  T^TT  Gramme. 

"  T^  Gramme. 


MEASURES  OF  EXTENSION. 
302.  The  Metre  is  the  unit  of  length. 

TABLE. 

10  Millimetres  =  1  Centimetre    =      .3937079  in. 

10  Centimetres  =  1  Decimetre     =    3.937079  in. 

10  Decimetres  =  1  Metre         =  39.37079  in. 

10  Metres  =  1  Dekametre    =  32.808992  ft. 

10  Dekametres  =  1  Hectometre  ■--=  19.927817  nl. 

10  Hectometres  ^  1  Kilometre     =       .62i:i824  mi. 

10  Kilometres  —  1  Myriametre  =    6.213824  mi. 

.W?,  Tin'  Are  U  tlio  unit  of  land  measure. 

TABLE. 

1  Centiare   ==  1  8q.  Metre      =      1.19r.034  sq.  yd. 
100  Centiares  =  \  Are  —  119.t)034  sq.  y«l. 

100  \tv6         =  1  Hectare         ^      2.47114  acres. 


•J n  I  1) KN OM I N  A T i:    N  U M 15 11116. 

*MH,  The  Square   Metre  is  the  unit  for  measuring 


100  Sq.  Millimetri  -  -j.  Centimetre  ==      .165 -{- sq.  in. 

100  Sq.  C'Cntimetres  =  1  Sq.  Decirpetre    =  15.5 -j-Bq.  in. 

100  ^v  Tv...:......,^    ~  1  ^'^,  }rHrr         -     ^VM^      =q.  yd. 

305,  The  Stere  is  the  unit  of  wood  and  solid  hk  umik  . 

TABLE. 

1  Dcciatere  ~  3.531  -f  cu.  ft. 
10  Decisteres  :=  1  Stere  —  35.316  +  cu.  ft. 
10  gterea         =  I  Deka«tere  =  13.079 +  cu   nI. 

306.  The  Cubic  Metre  is  the  unit  for  measuring  many 
ordinary  solids;  as  excavations,  embankments,  etc. 

T.VBLE. 

1000  Cu.  Millimetres  =  1  Cu.  Centimetre  =      .061  -f  cu.  in. 
1000  Cu.  Cenlimetrea  =  1  Cu.  Decimetre    =  61.026  cu.  in. 
1000  Cu.  Decimetres  =  1   Cu,  Metre,    -^  35.316  cu.  fi. 


MEASURES  OF  CAPACITY. 

307.  The  Litre  is  the  unit  of  capacity,  both  of  liquid 
and  dry  mcasiiro.     It  contains  about  a  quart,  liquid  mea.^ure. 

TABLE. 

10  Millilitres  =  1  Centilitre    =  .6102  cti.  in.  =  .338  fluid  oz. 

10  Centilitres  =  1  Decilitre     =  6.1022  cu.  in.  =  .845  gills. 

10  Decilitres  =  1  Litre        =  .908  quart  =  1.0567  qt. 

10  Litres  =  1  Dekalitre    =  9.08  quarts  —  2.6417  gal. 

10  Dekalitres  =  1  Hectolitre  =  2.8372 -fbu.  =  26.417  gal. 

10  Hectolitres  =  1  Kilolitre     =  28.372  +  bu.  =  264.17  gal. 

10  Kilolitres  =  1  Myrialitre  =  283.72  +  bu.  ^  2641.7  -f  gal. 


METRIC    SYSTEM. 


200 


MEASURES  OF  WEIGHT. 
308,  Tlic  Granime  is  the  unit  of  weight. 


10  Milligrammes         • 

10  Centigrammes 

10  Decigrammes 

10  Grammts 

10  Dekagrammes 

10  Hectogrammes 

](>  Kilogrammes 

10  Myriagrammes,  or) 

^00  Kilogrammes         j 

10  Quintals,  or  ) 

1000  Kilogrammes         j 


TABLE. 

1  Centigramme 
1  Decigramme 
1  Grinume 
1  Dekagramme 
1  Hectogramme 
(  Kilogramme,) 
^\     or  Kilo     j 
1  Myriagramme 

1  Quintal 

C  Tonnean.  or  ") 


.15432 +  gr. 
1.54324+  " 
15.43248+  " 

.35273  +  oz.  Av, 
3.52739+  "     '* 

2.20462+11.. 

22.04621+  "     " 

220.46212+  "     " 

2204.62125+  "     " 


The  Kifof/rainmCf  or  Kilo^  is  the  unit  of  common 
weight  in  trade,  and  in  a  little  less  than  2^  lb.  AvoirdujK^is. 

The  ToHltean  is  used  for  weighing  very  heavy  articles, 
aixl  !-  alx)ut  204J  lb.  more  than  a  ton. 


7'  V 


1.  What  niotnc  mut  (•()rns|)i.iMi>  mu>t  nearly  to  our  y 
How  many  metres  are  there  in  a  rod? 

2.  What  metric  measure  corresponds  most  nearly  to 
mile? 

3.  What  unit  expresses  nearly  one  ton? 

4.  What  unit  is  nearly  equal  to  one  quart? 

5.  He<luce  45  dekagrammes  to  gnunmes. 

6.  Express  7  dekametres,  25  centimetres,  as  metres. 

7.  How  many  litres  are   thcro  in  a  kilolitre*:'     In   a  <1 
litre? 


2()C)  DENOMINATE   NUMBERS. 

8.  ^V^lat  is  the  value  of  an  acre  in  metric  units? 

9.  Whnt  '""♦••!'•  ""'••-•M' -:.»>.wi^  ">"-t  nearly  to  one 

bushel? 

10.  IvedutHi  o6G.4ol  metres  to  tlucinietn  - 

11.  Whatwill  be  thecostof  n-^J  ir.  iw.,..,       .  ,. 

at  22  cents  per  hectogramme. 

12.  A  merchant  bought  38  gal.  of  wine  at  §2.15  jier  gal. 
Did  he  gain  or  !"-•■  ""1  1"'^^-  """-l^  1>y  s(*lliii2:  it  at  35  jx?r 
dekalitre? 

13.  Which  is  cheajier,  to  buy  cloth  at  $3  per  metre,  or  at 
$2.90  jxjr  yard?    How  much  cheaper? 

14.  How  much  carpet  a  metre  in  width,  is  required  to  car- 
pet a  room  4.2  metres  long  and  3.8  metres  wide? 

15.  How  long  must  a  pile  of  wood  be,  so  that  it  may  con- 
tain 12  steres,  if  it  is  3.5  metres  wide  and  3  metres  high? 

16.  How  much  is  gained  by  selling  a  piece  of  silk  100 
metres  in  length,  at  $2.25  j)er  yard,  if  it  cost  32  jxir  metre? 

17.  If  a  farm  contains  1400  ares,  what  will  be  its  value  at 
$2.50  \yeT  are? 

18.  A  barrel  of  flour  contains  196  pounds.  Express  its 
weight  in  metric  units. 

19.  How  many  hectares  are  there  in  a  farm  that  is  1000 
metres  long  and  180  metres  broad?  What  is  its  value  at 
3250  per  hectare ': 

20.  A  bin  is  5  metres  square  and  2.5  metres  high.  How 
many  hectolitres  of  wheat  will  it  hold ?        ..    ?  ^i      Q' 

21.  A  room  is  5.2  metres  long,  4.5  metres  wide,  and  3.2 
metres  high.  What  will  be  the  cost  of  plastering  it  at  35 
cents  per  square  metre? 

22.  Which  is  more  profitable,  and  how  much  per  ton,  to 
sell  sugar  at  11  cents  per  lb.  or  23  cents  per  kilo? 

23.  A  cask  holding  2  hectolitres  of  molasses  was  sold  at 
18|  cents  per  litre.  How  much  more  profitable  would  it  be 
to  sell  the  molasses  at  90  cents  per  gallon? 


:^^JlJjuu 


5 


PERCENTAGE 


309.  1.  In  a  quantity  of  sugar,  4  lb.  of  every  100  lb.  were 
wasted.     What  part  of  it  was  wasted? 

2.  A  laborer  digs  potatoes  for  10  bushels  out  of  every  100. 
"What  part  of  the  whole  does  he  get? 

3.  A  merchant  lost  83  out  of  every  $100  worth  of  goods 
soid,  on  account  of  bad  debts.  What  part  of  his  sales  did 
he  lose? 

4.  Millers  take  1  bushel  out  of  every  10  bushels  which 
they  grind  for  customers,  as  pay  for  grinding.  How  many 
hundredths  do  they  take? 

5.  In  a  company  of  soldiers,  1  out  of  every  4  men  was  killed. 
How  many  was  that  per  hundred,  or  jjei'  cent,  f 

6.  In  a  school,  5  out  of  20  pupils  are  more  than  14  years 
old.    How  many  is  that  per  hundred  ?    How  many  per  cent.  ? 

7.  A  man  sj^nt  83  out  of  every  84  earned.  How  many 
hundredths  of  his  money  did  he  s|>end?     What  per  cent.? 

"  A  man  whose  income  was  82500  annually,  saved  ^y^, 
<  1    10  jK-r  cent,  of  it.     How  many  dollars  did  he  save? 

!».  What  is  y^^,  or  5  per  cent.,  of  8800?  6  per  cent,  of 
8500? 

10.  What  is  2  per  cent,  of  8500?  4  ^x^r  cent,  of  8900? 
3  percent,  of  8500? 

11.  What  is  5  per  cent,  of  8^00?     «i  j^or  cent,  of  8900? 

12.  What  is  8  jxir  cent,  of  500  bushels?  10  i)er  cent,  of 
1000  pounds? 

(207) 


2(KS 


I'KUCENTAGK. 


DKilMT!'  '^  ^ 

310.  Per  Cent,  mcaiLs  by  t/LC  hiULdrt'l. 

It  Iti  a  contraction  of  the  Latin  ]ier  r^  ■' ■•  ■   >>•   i,.^  i.,..„ii,  V 

311.  The  Sifftt  of  Per  Cm  f.  .  Tbn-  -  m  a,! 
8  per  cent. 

31*2.  Pereetitlige  treats  of  computations  which  involve 
))er  cent. 

313.  iSinoe  jx>r  cent,  is  a  numlwr  of  hundredths,  it  is  usu- 
ally expressed  as  a  decimal.  It  mnv  also  l)o  fxprcs-sod  as  a 
common  fraction.     Thus, 

2  per  cent,  is  written      2<jfc,  .02,  or  jjj. 

5  per  cent,  is  written      5^,  .05,  or  jg^. 

47  per  cent,  is  written    47^,  .47,  or  ^^"j. 

135  i>er  cent,  is  written  \oo(fc,  1.35,  or  }^§. 

12i  per  cent,  is  written  12i^f,  .121,  or  |g^. 

J  per  cent,  is  written       |^,  .00^,  or  j^^. 

31  i  per  cent,  is  written  31^^<,  .31 1,  or  \\^. 

314:.  The  expressions  .12J,  .31  J,  etc.,  may  also  he  written 
.125,  .3125,  etc. ;  and  the  complex  fractional  forms  \^,  yjg^, 
etc.,  may  be  expressed  as  simple  fractions:  as,  ^\,  -^,  etc. 


Express  decimally 

,  and  in  the  smallest  terms  of  their  equiv- 

alent  common  frartions,  the  following: 

1.     10^. 

>'.     H%. 

17.  l^%. 

2.  12^%. 

10.     125^. 

.18.  20i%. 

3.     20%. 

11.     n%. 

19.      15?. 

4.     25%. 

12.     33^^. 

20.       1%. 

5.     30%. 

13.     \^%. 

21.     4i^. 

6.     75%. 

14.  i\n</c- 

22.    f^%. 

7.  871%. 

15.     Z\\</c. 

23.  7A%. 

8.     31%. 

IC.     G6§^. 

24.  374%. 

PERCENTAGE.  209 

Express  in  hundredths  or  jMjr  cent.,  ^  of  a  number;  ^  of 
^^;  i»  Aj  3sV;  ih'f  i*  t\5  Aj  "h'f  i'  s>  i'>  Jj  i?  +5  tVj 

"So!    f*    T^>    1S>    ZO'    77)- 

Problems  in  Percentage  involve  the  following  elements: 

315.  The  Sase  is  the  number  of  which  the  per  cent,  is 
taken. 

816.  The  Mate  is  the  number  of  hundredths  taken. 

317.  The  Pei*ceiitaffe  is  the  number  which  is  a  certain 
number  of  hundredths  of  the  base. 

318.  The   Amount  is  the  sum  of  the  base  and  per- 
centage. 

319.  The  Diffei'etice  is  the  base  less  the  percentage. 

In  the  formulas,  B.  representa  base ;  R.,  rate ;  P.,  percentage ;  A., 
amount;  and  D.,  difference. 

CASE  I. 

320.  To  find  the  IVrceiitat^e  when  the  Base  anil  Rate 
are  given. 

EXERCISES, 

1.  What  is    10  per  cent.,  or  -ji^,  of  $150? 

2.  Wluit  is      5  i)er  cent.,  or  y^,  of  3400? 

8.  What  is    20  per  cent.,  or  -,2^,  of  300  bu.? 

4.  What  is  12i  per  cent.,  or  ||^,  of  800  gal? 

5.  What  is    15  per  cent.,  or  ^,  of  $400? 

6.  What  is  33J  per  cent,  of  $600? 

7.  Wlmt  is  371  per  cent,  of  160  men? 

8.  Wliat  is    40  jrt  cent,  of  200  tons? 

9.  What  is       I  I.  ,  cent,  of  800  horses? 

10.  What  is      T)  pt  r  cent,  of  700  i)upils? 

11.  What  is    25  iwr  cent,  of  124  yards? 

14 


210  PERCENTAGE. 

12.  What  is  Q\  per  cent,  of  «32.64? 

PROCESS. 

$32.64  Xi^jT       ^--^^  ANALVbis.— Since    6}^    of    an/ 

number  is  /g^,  or  ^\,  of  it,  6\fc  of 
Ory  $32.04  is  /,.  of  $32.64,  which  is  $2.04. 

$32.64x.06i-=82.04     ^'         ,_    , 

*  Since  e^jii,  of  any  number  is  M\ 

of  that  number,  G^ji^o  of  $32.G4  is 
FORMULA.  ^^  ^j  ^2.64,  which  is  $2.04. 

B  X  /?  =-  P 

liuLE. — Multiply  Oit  buse  by  the  rate. 


13.  Find  35%  of  $21.75. 

14.  Find  48%  of  $13.42. 


15.  Find  33J%  of  465  gal. 

16.  Find37J%of816mi. 


17.  A  fariiKr  who  had  a  flock  of  450  sheep,  sold  33J% 
of  them.     How  many  had  he  left? 

18.  A  man  whose  salary  was  §2000  per  year,  spent  85% 
of  it.     How  much  did  he  save  annually? 

19.  A  farmer  sold  37J%  of  his  crop  of  816  bu.  of  wheat 
at  SI. 56  per  bu.,  and  the  rest  at  81.60.  How  much  did  he 
realize  from  the  sale  of  liis  wheat  ? 

20.  If  a  merchant  makes  a  deduction  of  5%  from  a  bill 
of  S318.57,  how  much  must  be  paid  him? 

21.  A  man  bought  a  farm  for  830000,  and  sold  it  for  a 
gain  of  25%.     How  much  did  he  get  for  it? 

22.  Mr.  Seymour  sold  83000  worth  of  flour  at  a  loss  of 
'i2}r%.     How  much  did  he  realize  from  the  sale? 

23.  A  man  having  840000,  invested  15%  in  bank  stock, 
27%  of  it  in  bonds  and  mortgages,  and  the  rest  in  a  flouring 
mill.     How  much  did  the  mill  cost? 

24.  Two  brothers  each  inherited  818500.  The  elder  in- 
creased his  inheritance  8%  per  year  for  3  years.  The  younger 
lost  33J%  of  his  in  the  same  time.  What  was  then  the  value 
of  the  inheritance  of  each  ? 


PERCENTAGE.  211 

CASE  II. 

321.  To  flu<l  the  Rate  when  the  Base  and  Percentage 
are  given. 

1.  If  a  man  earn  §100  and  spend  $50  of  it,  what  part  of 
it  does  he  spend  ?  How  many  hundredths  ?  How  many  per 
cent.  ? 

2.  In  a  piece  of  cloth  containing  36  yards,  9  yards  were 
damaged.  What  part  of  it  was  damnged?  How  many  liun- 
dredths  of  it?     How  many  per  cent.? 

3.  When  I  spend  \  of  my  money,  how  many  hundredths 
of  it  do  I  spend?     How  many  per  cent.? 

4.  If  a  farmer  loses  ^  of  his  crop  by  a  flood,  how  many 
hundredths  of  it  does  he  lose  ?    How  many  per  cent.  ? 

5.  If  a  merchant  sells  i  of  his  goods  annually,  how  many 
hundredths  does  he  sell  ?    How  many  per  cent.  ? 

6.  A  farmer  had  25  sheep,  and  10  of  them  died.  What 
part  of  his  sheep  died  ?    What  per  cent,  of  them  ? 

7.  What  part  of  §15  are  §3?     What  per  cent.? 

8.  What  part  of  12  bushels  are  G  bushels?  What  per 
cent? 

9.  What  per  cent,  of  24  cows  are  8  cows?  What  per 
cent,  are  12  cows? 

10.  What  per  cent,  of  200  students  are  40  students?  Are 
60  students? 

11.  What  per  cent  of  150  acres  are  30  acres?  Are  50 
acres?    Are  75  acres? 

12.  What  jxjr  cent  of  80  hours  are  16  hours?  Arc  20 
hours?    Are  40  hours? 

13.  What  |)er  cent  of  90  gallons  are  30  gallons/  Are  00 
gallons?    Are  45  gallons? 

14.  If  a  man  who  earns  860  per  month,  expends  840  yter 
month  for  necessary  (.\|"'>.«:*-:  what  jx»r  cent,  of  his  earnings 
does  he  save  ? 


212  PERCENTAGE. 

15.  A  merchant  having  375  yards  of  cloth,  sold  150  yards 
of  it.     What  per  cent,  did  he  sell? 

PROCESS.  Anm.vsis. — loO  yards 

150  yd.  =  W  of  375  yd.,  or  "'    -   ,  or  ^  of  375 yards. 

i  of  375  yd.,  or  40$^  of  375  yd.       dr<^UiB,  ,quaU  10  hun- 

Of  dredths;     therefore    150 

yards  are  .40,  or  40o^, 
150  yd.  --375  yd.  =  .40,  or  40%         of  375  yards.    Or,       ' 

Since  the  percentage  is 
FORMULA.  a  product  of  the  base  by 

p.^  B  =  R.  *^®  *'^^^»  ^^  ^'^  divide  the 

percentage  by  the  base  we 
shall  obtain  the  rate.  Therefore  we  divide  150  by  375,  and  obtain  for 
a  quotient  .40.  or  lOr;. 

Rule. Ihmir  im   ^nn(  itiniit    (til  the  boSC. 

16.  AVhat  per  cent,  of  360  men  are  60  men? 

17.  What  per  cent,  of  840  men  are  360  men? 

18.  What  per  cent,  of  380  pages  are  120  pages? 

19.  What  per  cent,  of  45  hours  arc  25  hours? 

20.  What  per  cent,  of  50  yards  arc  27  yards? 

21.  What  per  cent,  of  36  pounds  are  24  pounds? 

22.  A  former  who  had  a  farm  of  540  acres,  sold  210  acres 
of  it.  What  per  cent,  of  it  did  he  sell? 

-  23.  A  man  whose  annual  income  is  81800,  spends  §1600  of 
it.  What  i^er  cent,  of  it  does  he  spend?  What  per  cent,  of 
it  has  he  lefl? 

—  24.  A  grocer  sold  tea  for  81  that  cost  him  8  .75.  What 
per  cent,  of  the  cost  did  he  gain? 

25.  What  per  cent,  of  30000  bushels  are  50  bushels? 

26.  What  per  cent,  of  the  cost  does  a  hatter  gain  by  sell- 
ing hats  at  87  each,  that  cost  85.50? 

27.  A  real-estate  agent  gets  860  for  selling  my  house  for 
84000.     What  %  of  the  sale  does  he  receive  for  his  services? 


I'hlii  i.MAGE.  213 

28.  I  paid  ^^^2o  l()r  insuring  a  l)oat-load  of  wheat  valued 
at  SIPOOO.  Wliat  %  of  the  value  of  the  cargo  was  received 
for  insuring  it? 

29.  A  man  who  had  1000  acres  of  land,  gave  \  of  it  to  his 
eldest  son,  ^  of  it  to  another,  and  the  remainder  he  divided 
equally  hetween  his  3  daughters.  What  %  of  the  whole  did 
each  receive? 

CASE  III. 

322.  To  fiii«l  the  BiiHe  when  (lie  Percentage  and  Rate 
are  given. 

1.  A  man  spent  $15,  which  was  10%  or  -^j^  of  all  the 
money  he  had.     How  much  money  had  he? 

2.  My  net  profit  from  an  investment  was  8800,  which  was 
25*%  or  ^Q*Q  of  the  amount  invested.  Hf)W  much  had  I  in- 
vested ? 

3.  Of  what  ^UIll  is  18  dollars  o^)./^  ,  oi  ^^^y,  or  J? 

4.  Of  how  many  days  are  15  days  20%  ?    30  days  37^%  ? 

5.  Of  what  sum  is  25  dollars  62  J%,  or  f  J^,  or  |? 

6.  Of  what  number  is  120    6%?    150,30%?    180,    60%? 

7.  Of  what  number  is    40  80%  ?      20,  60%  ?      30,  150%  ? 

8.  A  drover  lost  450  sheep,  which  was  75%*  of  his  flock. 
How  many  sheep  had  he? 

I  !  Analysis.— Since  75^  or  ^^ff  o'  i©^  the 

tVW  or  ^  ^^450     "umher  is  4oO,  \  of  (he  number  h  J  of  450 

1 1  5  0     ^^  ^'^'   ""*'  f^^nce  150  is  |  of  the  numl)cj;, 

,,^.    ,^  _   p(\(\     ^^^^  whole  number  of  sheep  will  be  4  times 
>\hole-^bUU     j5^^  ^^  QQQ     Therefore  he   had   (KK)  t^heop. 
Or,  Or, 

4r>n  •     75-600         ^'"^   ^''^   pcrcentngf    i>    ilu    pi.Klurt    «.i 
the  base  by  the   rate,  if  the   percentage  is 
divided  hv  the  rate  the  quotient  will  be  the 
base. 
P  :    R  -:  B.  TherticiL  wc  divide  450  by  .75. 

!  -Divide  the  percentage  by  the  rate. 


214  PERCENTAGE. 


Of  what  number  is 
9.    385   12J^%? 

10.  245     10%  ? 

11.  125     15%? 
]-'    7  1.  ;)3i%? 


Of  what  number  is 

13.  $53.25    10%? 

14.  27.5  bu.   8%? 

15.  168  men  8%? 

16.  231  oxen  7%? 


--17.  A  farmer  sold  275  barrels  of  apples,  which  were  75% 
of  all  he  had.     How  many  had  he? 

18.  A  man  sold  25%  of  a  mill  f  At  this  rate 
what  was  the  mill  worth  ? 

19.  A  man  who  owned  40%  of  a  foundry  sold  25%  of  his 
share  for  810000.     What  was  the  value  of  the  foundry? 

20.  A  farmer  after  selling  110  A.  43  sq.  rd.  of  land  had 
90%  of  his  land  left.     How  much  land  had  he  at  first? 

21.  A  farm  cost  S3000.  One-third  of  this  sum  was  62J% 
of  what  the  house  and  barn  on  the  farm  cost.  What  was  the 
cost  of  the  house  and  barn? 

22.  A  man  indebted  to  me  })aitl  me  880,  which  was  8^% 
of  ^  the  amount  due.     How  much  did  he  still  owe? 

23.  A  merchant  sold  4500  bushels  of  wheat  at  81.60  per 
bushel.  The  amount  received  was  90%  of  ti"^  rost  of  the 
wheat.     How  much  did  it  cost? 

24.  Mr.  A.  sold  a  lot  for  88000,  which  was  only  40%  of 
the  amount  he  paid  for  it.     How  much  did  he  pay  for  it  ? 

25.  A  man  pays  8600  a  year  rent;  75%  of  this  sum  is 
jiist  33^%  of  4^  his  income.     What  is  his  income? 

-    26.  A  man  owning  ^  of  a  vessel  sold  25%  of  his  share  for 
83350.50.     At  that  rate  what  was  the  value  of  the  vessel? 

27.  The  amount  paid  by  insurance  companies  to  the  people 
of  St.  John,  New  Brunswick,  for  losses  caused  by  the  great 
fire  in  1877,  was  about  87500000,  which  was  37J%  of  the 
estimated  loss.     What  was  the  estimated  loss? 

28.  25%  of  i  of  60  is  75%  of  i  of  what  number? 

29.  i  of  40%  of  100  is  5%  of  10  times  ^  of  what  number? 


PERCENTAGE.  215 


CASfe  IV. 

323.  To  find  flic  Base  when  the  Amount  and  Rate 
are  given. 

1.  A  gentleman  increased  his  collection  of  horses  by  an 
addition  of  J  of  the  number,  and  then  he  had  15.  How 
many  had  he  before  he  made  the  addition  ? 

2.  A  coal  dealer  in  selling  coal  at  86  a  ton  received  20% 
or  -J  more  than  it  cost  him.     What  did  it  cost  him  ? 

3.  A  grocer  in  selling  sugar  at  $.11  a  ix)und  gains  10% 
or  -^  of  the  cost.     What  was  the  cost? 

4.  A  merchant  sold  cloth  at  an  advance  of  25%  on  the 
cost,  receiving  §1.25  per  yard  for  it.     What  was  the  cost? 

*  5.  A  man's  monthly  expenses  were  33J%  more  during  1876 
than  during  1875.  During  1876  they  were  $120.  What  were 
they  in  1875? 

6.  A  certain  number  increased  by  20%  of  itself  is  36. 
What  is  the  number? 

7.  After  adding  to  a  number  37^%  of  it,  the  sum  is  33. 
AVhat  is  the  iiumlx^r? 

8.  What  number  increased  by  35%  of  itself  equals  540? 

PROCESS.  Analysis. — Since  the  number  is  in- 

I  as  __  5  ^ Q  creased  by  35o{^, or /j'^j  of  itself, the  amount 

J     A  will  be  l^j^  or  \^^  times  the  numljer: 

TOO  nnA      o!n/M>     13  5     e\(     iV\r,     nimtlv.'t..    f^Afi  1 


Tlie  number  =  4 00 


and  since  jJJ  of  the  number  =  540,  yj^ 
of  it  =  j\-g  of  540,  which  is  4;  and  since 
Or,  4  is  yjj  of  the  number,  the  numUr  will 

M  4-   35)  ^  ^^  times  4,  which  is  400.     Or, 

JAA  "^  number  increased  by  Zb^c  of  itself 

540-^1.35  =  400  equals  135%  or  1.35  of  itself.  And  since 
1.35  timefl  the  number  equals  540,  the 
number  may  be  found  by  dividing"540 


FORMULA. 


A-^(l-\   R)  =  B.      by  1.35. 

Rule. — Divide  the  atnount  by  l-\-Vie  rate. 


216  PERCENTAGE. 

9.  What  number  increased  by    27%  of  itself  equals  508? 

10.  What  numl)er  increased  by  33^%  of  itself  equals  492? 

11.  What  nunil)er  increased  by  16§%  of  itself  equals  329? 

12.  What  number  increased  by  62^%  of  itself  equals  910? 

13.  A  man  owes  $15400,  which  is  10%  more  than  his 
property  is  worth.     What  is  the  value  of  his  property  ? 

14.  A  man  sold  a  horse  for  $345,  which  was  15%  more 
than  it  cost  him.     How  much  did  it  cost? 

15.  A  clerk's  salary  was  increased  30%,  and  now  it  is 
$1950.     What  was  it  before  the  increase? 

--16.  A  man  expended  $3750  in  repairs  upon  his  house.  This 
sum  was  25%  more  than  -J  the  cost  of  the  house.  How 
much  did  it  cost? 

17.  The  number  of  pupils  in  a  certain  school  durin;]^  1876 
was  872,  which  was  9%  more  than  the  number  in  attendance 
during  1875.     What  was  the  attendance  during  1875? 

CASE  V. 

324.  To  find  the  Base  when  the  DiO'erence  and  Rate 
are  given. 

1.  A  gentleman  sold  25%  or  ^  of  the  number  of  his  horses 
and  had  15  left.     How  many  had  he  ? 

2.  By  selling  coal  at  $6  per  ton  a  coal  dealer  lost  20%  or 
\  of  the  cost.     What  was  the  cost  ? 

3.  A  grocer  sold  sugar  at  9c.  per  pound,  and  lost  10%  or 
^  of  the  cost.     What  did  it  cost  ? 

4.  A  merchant  sold  cloth  at  $  .75  per  yard,  thereby  losing 
25%  of  the  cost.     What  was  the  cost? 

5.  A  man's  monthly  exi^enses  are  33^%  less  this  year  than 
last  year.     This  year  they  are  $120 ;  what  were  they  last  year  ? 

6.  A  certain  number  diminished  by  20%  of  itself  is  36. 
What  is  the  number? 

7.  What  number  dimmished  by  10%  of  itself  equals  45? 


peu(^knta(;k.  ^17 

8.  After  .suhtrucliiii^  IVuiu  ii  iiunibor  o7^%  <»i'  it,  the  re- 
mainder is  25.     What  is  the  number? 

9.  What  number  diminished  by  27%  of  itself  equals  401.5? 

PROCESS.  Analysis. — Since  the  number  is  de- 

78  =401.5  creased  by  27f/r,  or  ^<^\  of  itself,  the 

^  ^^ _*  _  remainder  will  be  -j^^  of  the  number, 

*  *o  *  which  equals  401.5;  yjj  of  it  equals 

The  number  =  550  7V  of  401.5,  which  is  5.5;   and  since 

^  5.5  is  y^^  of  the  number,  the  number 

^'*'  will  be  100  times  5.5,  which -is  550. 

(1  — .27)  Or, 

401    5-i-    73=:550  ^  number  diminished  by  27 <^   of 

itself,  equals  73c{:,  or  .73,  of   itself; 

and  since  .73  of  the  number  equals 

FORMULA.  .^Qj  5^   ^^^  number  will  be  equal   to 

/'  l  —  R)  =  B,         401.5 ^ .73,  which  is  550. 

Rule. — Divide  the  difference  by  1  minus  the  rate. 

10.  What  number  diminished  by  36%  of  itself  equals  336? 

11.  What  number  diminished  by  40%  of  itself  equals  432? 

12.  What  number  diminished  by  55%  of  itself  equals  285? 

13.  What  number  diminished  by  28%  of  itself  equals  307? 

14.  A  clerk,  after  paying  out  75%  of  his  salary,  had  S450 
lef^.     What  was  his  salary  ? 

15.  A  farmer,  after  selling  30%  of  his  wheat,  found  that 
he  had  350  bushels  left.     How  much  had  he  at  first? 

16.  A  man  sold  some  land  for  30%  less  than  he  asked  for 
it,  getting  $29.24  per  acre..    What  was  his  askmg  price? 

17.  A  regiment  losinir  l"'^-  "^  lf<  m.Mi  hnA  .'")27  h^ft.  TTow 
many  had  it  at  first? 

18.  A  8|)oculator  lost  lU^;,  ot  ins  money  during  the  year 
1875  and  10%  of  the  remainder  during  1876.  Ho  then  had 
$40500  left.     How  much  bad  he  at  first? 

19.  A  merchant's  profit  in  1876  was  $l()ol^,  whkU  \\a.H 
23%  le86  than  in  1875.     What  was  his  profit  in  1875? 


i 


V 


Hi 


ST 


325.  1.  When  a  sum  equal  to  6%  of  the  amount  of 
money  lent  is  paid  for  the  use  of  it  for  one  year,  how 
much  will  be  paid  for  the  use  of  SI 00  for  1  year?  For  2 
years? 

2.  When  the  allowance  for  the  use  of  money  is  6%  per 
year,  what  is  the  allowance  for  the  use  of  8100  for  1  year? 
For  2  years?     For  3  years?     For  3J  years? 

3.  When  the  sum  paid  for  the  use  of  money  is  8%  per 
year,  what  must  be  paid  per  year  for  $50?    For  3500? 

4.  When  the  sum  paid  for  the  use  of  money  is  12^  yearly, 
what  must  be  paid  for  the  use  of  SlOO  for  1  year? 

5.  When  the  allowance  for  the  use  of  money  is  8%  per 
year,  what  must  be  paid  for  the  use  of  $100  for  6  months? 
For  1  month?  For  ^  month?  For  \  month?  For  i  month? 
For  10  days?     For  20  days? 

6.  When  6^  is  paid  per  year  for  the  use  of  money,  how 
much  will  $500  amount  to  in  2  years?     In  3  years? 

7.  When  $500  is  loaned  for  1^  years  at  8%  per  year,  what 
will  be  the  amount? 

DEFINITIONS. 

326.  Interest  is  the  sum  paid  for  the  use  of  money. 

327.  The  Principal  is  the  sum  for  the  use  of  which 
interest  is  paid. 

(218) 


INTEREST. 


219 


328.  Tlie  Amount  is  the  sum  of  the  principal  and  in- 
terest. 

'329.  The  Hate  of  Interest  is  the  annual  rate  per 
cent. 

330.  Legal  Interest  is  interest  according  to  rate  fixed 
by  law. 

331.  Usuf*y  is  interest  computed  at  a  higher  rate  than 
the  law  allows. 

332.  A  Note^  or  Promissory  Note,  is  a  written 
promise  to  pay  a  sum  of  money  at  a  given  time. 

333.  Principle. — The  interest  is  equal  to  Hie  product  of  Hie 
pniticipalj  rate,  and  time  expressed  as  years. 

334.  When  the  rate  per  cent,  is  hot  specified  in  notes, 
accounts,  etc.,  the  legal  rate  is  always  understood. 

On  debts  due  the  United  States  the  rate  is  6^. 

The  following  table  contains  the  rates  of  interest  in  the 
United  States.  The  first  column  gives  the  legal  rate,  the 
second  the  rate  that  may  be  collected  if  agreed  to  in  writing. 


NAME   OF  STATE. 

Rati 
Pm  C«nt. 

NAME   OF  STATE. 

Rate 

Pt»    CtKT. 

10 
10 

10 

'"io" 

Any. 
Any. 

AllT. 

^';^• 

"io' 

A„.. 

S4 

8 
8 

"io" 

u 
...... 

Aoy 

^v 

10 

iJ 

10 

10 

Any. 
AdV." 

C.lirornU 

Coaac«tlcut. 

Colorado 

New  llam|.tbir« 

S'-fr  Jvncx 

New   MfXico 

North  Carollita. 

8 

10 

Florid*    

Nevada 

'T 

Ohio 

Orpfon 

»*.-oni>yl»iiiil». , 

KhodcUland 

South  CaroIlD*. 

idahJi.::::::.:::::::::.: : 

iniiH'to .' 

ludiau 

ladlBU  Territory.. 

Iowa 

«•■«•. 

f«S5V.' :••:•::• 

It 

AiV." 
ADjr. 

"ii" 

ABjr. 

Tf>nDe»ae« 

Texa. 

rt«h ; 

Vermont. 

Vlrjji.la 

Wmi  VIrplnl 

!»■»■• 

MarylMd 

V/- 

SiBBeUi.:::::::::::::::::::::;:::::: 

AV. 

22Q  PERCENTAGE. 


TO  CX)MPUTE  INTEREST. 

335.   1.  What  is  the  interest  of  S200  for  1  year  at  6^? 

2.  What  is  the  interest  of  $200  for  2  years     at  7^  t 

3.  What  is  the  interest  of  $300  for  3  years     at  5^  ? 

4.  What  is  the  interest  of  $400  for  IJ  years   at  8^? 

5.  What  is  the  interest  of  $400  for  3  months  at  6^  ? 

6.  What  is  the  interest  of  $600  for  1  month   at  6^  ? 

7.  What  is  the  interest  of  $600  for  10  days    at  6^  ? 

8.  What  is  tlie  interest  of  $500  for  15  days    at  8^  ? 

9.  What  will  be  the  amount  of  $100  for  2    years  at  6^  ? 

10.  What  will  he  the  amount  of  $200  for  3    years  at  4%  ? 

11.  What  will  be  the  amount  of  $300  for  2^  years  at  efc  ? 

12.  -What  will  be  the  amount  of  $150  loaned  for  IJ  years 
at  bfc^ 

13.  Fin(ltheinterestof$284.27for2yr.  7mo.  12  da.  at  6^? 

I'ROCEss.  Analysis. — Since  the  in- 

•  o  9  g  i    2  7  tereat   for   1   year    is  6^   of 

'  f^n  the  principal,  we  find  .06  of 

•  $234.27,  which    is  $14.0562; 


$14.0562  Int.  for  1  yr.  and  since  $14.0562  is  the  in- 

2  terest  for  1  year,  the  interest 

$28.1124  Int.  for2yr.  ^o""  -  y^^^  ^'^'1  ^  ^wice  that 

Q    o  aT  a  r  .  t    ^f^TL  sum,  which  is  $28.1124.    The 
O .  U  O  7  O  Int.  for  7  2-5  mo.  .       '        ,       ,  ,     . 
interest  for  1  month   is  one- 

$36.78         Int.  2  yr.  7 mo.  12  da.  twelftli  of  the  interest  for  1 

year,  or  $1.1713 ;  and  the  in- 


Or, 

!7 
.06 


terest  for  7  mo.  and  12  days, 

8  2^427  ^^  ^^  "*^*'  ^^  '^^  times  $1.1713, 

or  $8.6676.  This  added  to  the 
interest  for  2  years,  gives  the 


12)$14.0562  Int.  for  1  jrr.  interest  for  2  years,  7  months, 

$1.1713  intforlnio.  ^^  days.     Or, 

fy  ^     .  We   may  find  the  interest 

'—  for  1  year  as  before,  and  then 

$36.78  Int.  2  yr.  7  mo.  12  da.  for  1  montli.     We  then  mul- 


lliTEREST.  221 

tiply  the  interest  for  1  month  by  the  number  of  months  and  fractions 
of  a  month.  Thus,  in  2  years,  7  nionthn,  there  are  31  months,  and  in 
12  days  there  are  J  J  or  y*(y  of  a  month. 

Therefore,  the  entire  interest  may  be  found  by  multiplying  $1.1713 
by  31.4,  which  is  $30.78. 

Since  there  are  30  days  in  a  month,  one-third  of  the  number  of 
days  will  be  tenths  of  a  month. 

Rule. — I.  Find  the  interest  for  1  year  and  multiply  this  by 
the.  time  expressed  as  years  and  fractions  of  a  year.     Or, 

II.  Find  tlie  interest  for  I  month  and  multiply  this  by  Vie  time 
expressed  as  months  and  fractions  of  a  month. 

14.  What  is  the  interest  of  $25.16  for  1  yr.  6  mo.  at  6^  ? 

15.  What  is  the  interest  of  $36.24  for  2  yr.  4  mo.  at  7%  ? 

16.  What  is  the  interest  of  $48.20  for  2  yr.  4  mo.  at  8^  ? 

17.  What  is  the  interest  of  $2000  for  3  yr.  7  mo.  at  9^  ? 
„18.  Find  the  amount  of  $590.50  for  '3  yr.  6  mo.  at    7%. 

19.  Find  the  amount  of  $640.82  for  2  yr.  7  mo.  at    8^. 

20.  Find  the  amount  of  $725.83  for  3  yr.  6  mo.  at  10^. 

21.  Find  tlie  amount  of  $618.24  for  2  yr.  5  mo.  at    Sfc 

22.  Find  the  amount  of  $312.29  for  3  yr.  5- mo.  at    6^. 

23.  Find  the  interest  of  $718.24  for  5  mo.  10  da.  at    dfc 

24.  Find  the  interest  of  $127.46  for  3  mo.  15  da.  at    7fc' 

25.  Find  the  interest  of  $364.18  for  2  mo.  12  da.  at  8^^. 

26.  Find  the  interest  of  $318.29  for  9  mo.  10  da.  at  7}%. 

27.  Find  the  interest  of  $312.24  for  2  mo.  20  da.  at  8^. 
"28.  Find  the  interest  of  $1614.25  for  20  da.  at      7^. 

29.  Find  the  interest  of  $1318.29  for  24  da.  at    10^. 

30.  Find  the  interest  of  $4684.68  for  11  da.  at  V2ifo. 

31.  If  you  lend  $500,  how  much  will  be  due  you  in  3  yr. 
6  mo.  21  da.,  interest  at  7^  ? 

32.  What  is  the  interest  on  $784.25  from  Au^.  7,  1874,  to 
July  19,  1877,  at  Sfc  ?     What  is  the  amount? 

33.  How  much  interest  isdue  on -$500,  that  ha.-  Ir  i  .  .m  d 
at  interest  sinee  Jan.  1,  1876? 


222  PERCENTACiE. 


OTHER  METHODS. 

ALIQUOT  PAT?T^ 

3iJ6,    1.  What  is  the  interest  an  .    mm  in,,  -u     .._  ...,j  jor 
2  yr.  5  mo.  15  da.  at  7^  ? 

PROCESS.  Analysis.— Since   the  in. 

$520.32  terest  is  7</o  of  the  principal, 

.07  we  find  .07  of  $.520.32,  whicli 

is  $36.4224,  the  interest  for  1 
year.  Twice  $.30.4224  give* 
the  interest  for  2  years,  which 

$72.8448 -yr.  is  $72.8448.    One-thinl  of  the 

12.1408  Int.  for  4  ma  interest  for  1  year  is  $12.1408. 

3.0352  Int  fori  mo.  ^''^    ^"^^^^^    ^^'^    ^    °^«"^^>«- 

One-fourth  of  the  interest  for 


$36.4224  Int.  fori  yr. 


1 .  5 1  7  6  Int  for  15  da. 


4  months  is  $3.0.3.'32,  the  in- 

$    89.5384  Int  for  2 yr.  5 mo.  15 da.      terest  for  1  month.    One-lialf 

$520.32         Principal.  the   interest  for   1    month    is 

oaf\a   aa         »  »  $IM76.   the   interest   for   15 

$609.86         Amount  ^  '     ,  .      . 

days.      The    sum    of    these 

amounts  is  the  interest  for  2  years  5  months  15  days,  which  is  $89.5384. 

This  sum,  added  to  the  principal,  gives  the  amount. 

Solve  the  following  by  aliquot  parts: 

2.  What  is  the  interest  of  $324.22  for  3  yr.  4  mo.  at  6^  ? 

3.  A\Tiat  is  the  interest  of  8218.90  for  2  yr.  7  mo.  at  7^  ? 

4.  What  is  the  interest  of    $36.48  for  2  yr.  5  mo.   15  da. 
at  6;^? 

5.  What  is  the  interest  of    840.28  for  1  yr.  7  mo.  20  da. 
at6j^? 

6.  What  is  the  interest  of    856.24  for  2  yr.  5  mo.  18  da. 
at  7;^? 

7.  What  is  the  interest  of    $24  96  for  3  yr.  1  mo.     6  da. 
at  8^? 


INTEREST.  223 

8.  What  is  the  interest  of  $48.72  for  2  yr.  2  mo.  16  da. 
at  8^? 

9.  What  is  the  interest  of  SSO.LS  hn-  'l  yr.  :>  mo.  21  da. 
at  10^? 

10.  What  is  the  interest  of  $20.25  for  3  yr.  1  mo.  16  da. 
atl2i^? 

11.  What  is  the  interest  of  $30.24  for  2  yr.  8  mo.  15  da. 
at  8^? 

SIX  PER  CENT.  METHOD. 

337.  The  interest  on  $1,  at  6^  per  annum, 

For  12  months,  is 06 

For    2  months,  is    .       .       .       .       .       .01 

For    1  month  (30  days),  is.       .       .       .005 

For    6  days      Q  month),  is.       .       .       .001 

For    1  day,  is OOOJ 

1.  What  is  the  interest  of  $125  for  2  yr.  3  mo.  16  da.  at  6^  ? 
PROCESS.  Analysis.— Int.  of  $1  for    2  yr.  at  6^  is  .12. 

^  ^  ^  ^  Int.  of  $1  for  16  da.  at  (^<fo  is  .002J. 

«^  The  sura  of  these,  .137  J,  is  the  interest  of  $1  for 

$17,208  *he  given  time  at  the  given  rate,  and  since  the 

interest  of  $1  is  .137|,  the  interest  of  $125  will  be 

125  timcfl  that  sum,  which  is  $17,208. 

1.  Wlien  it  is  required  to  find  the  interest  at  any  other  rate  tlian 
6^,  first  find  it  at  6^,  then  increase  or  decrease  this  result  by  such  a 
part  of  it  as  the  given  rate  is  greater  or  less  than  6^.  Thus,  if  the 
rate  is  l(fc,  increase  the  interest  at  6^  by  J  of  it;  if  the  rate  is  h^c, 
decrease  it  by  J  of  it;  if  the  rate  is  8<^,  increase  if  l»v  -.  ..r  '  of  it ; 
if  the  rale  is  9^,  increase  it  by  \  of  it,  etc, 

2.  Hxact  or  Accurate  interest  requires  tlua  lii.  ^  u  m.-mhu  i^ 
considered  365  days,  for  a  common  year,  and  366  days  for  a  leap  year, 
instead  of  the  ordinary  method  of  conndering  12  months  of  30  days 
each,  or  360  days  a  year. 


224  PERCENTAGE. 

Find  the  interest  on  the  following: 

2.  On  a  not(                '..2(5  for  1  yr.  4  mo.  13  da.  at  G^  ? 

3.  On  a  note  i...  •  .m„s.18  for  3  yr.  5  mo.  22  da.  at  Cy^^,  ? 

4.  On  a  note  for  $284.25  for  2  yr.  7  mo.  1<S  da.  at  6^? 
T).  ( )ii  a  note  for  8183.17  for  1  yr.  8  mo.  17  da.  at  6^  ? 
<;,  On  a  note  for  $215.25  for  3  yr.  2  mo.  18  da.  at  6j^? 
7.  On  a  note  for  $204.37  for  2  yr.  5  mo.  15  da.  at  5^  ? 

_  8.  On  n  S18G.15  for  3  yr.  7  mo.  23  da.  at    7^? 

9    n.,  ..  , ,.,,  S315.30  for  1  yr.  9  mo.  27  da.  at    8^? 

1  note  for  $379.15  for  1  yr.  8  mo.  11  da.  at    6^? 

11.  On  a  note  for  $685.31  for  4  yr.  1  mo.  15  da.  at    7^?. 
Ti.  On  a  note  for  $516.28  for  3  yr.  6  mo.  28  da.  at    5^? 
J  ;.  <  )n  a  note  for  $423.15  for  2  yr.  7  mo.  10  da.  at    Qfo  ? 
14.  ( )n  a  note  for  $304.27  for  1  yr.  3  mo.  21  da.  at  10^;? 
ir,.   (  )n  .,  n..f..  f^.r  S.->ir,.24  f^.r  2  yr.  1  mo.  13  da.  at  12%? 

I'iiid  the  amount  <'t'  the  Ibliowiiij  \\'>U'<  when  due: 
It).   $150.15.  Cincinnati,  0.,  Jan.  31, 1877. 

Three  months  after  daley  for  value  received,  I  promise  to  pay 
Jofm  T.  Jones,  or  order,  One  Hundred  Fifty  -^^  Dollars,  with 
interest  at  6;?.  Charle-s  C.  Thomson. 

17.  $32S.35.  St.  Paul,  Minn.,  Oct.  1,  1^77. 
0/^  ///'■  l~)fli  <l  I  >i  of  January,  iS7S,  for  value  received,  I  promise 

to  paij  in  N.  E.  Ihnjt,    or  order.  Three  Hundred  Twenty-Eujht 
Tih  ^^"^^'^'"S  ^*^^  hvterest  at  S%.  J    W    Ray 

18.  $315.75.  Leavenworth,  Kan.,  May  ;j,  Js7G. 

For  value  reeeired,  on  demand  I  promise  to  pay  to  J.  C.  Coe, 
or  order,  Three  Hundred  Fifteen  j\f^  Dollars,  ivitJi  interest. 

Henry  B.  Robeson. 
Paid,  Juno  5th,  1.^77.     How  mncli  wa.-^  due? 


I.NTERiST.  225 


nOAfPOUND   INTEREST. 

338.  ('iPtnj)otniff  Interest  is  interest  upon  the  prin- 
cij)al  and  its  unpaid  interest,  combined  at  regular  intervals. 

It  is  usually  compounded  annually,  semi-annually,  or  quarterly. 
Unless  some  other  condition  is  mentioned  in  the  written  obligation, 
the  interest  is  understood  to  be  compounded  annually. 

WJiITT£N    EXERCISES. 

1.  Find  the  compound  interest  of  $250  for  2  yr.  3  mo.  at  6j!^. 

PROCESS.  Analysis.— Since  the  in- 

i250  Priu.  for  1st  yr.  Merest     is    compounded    an- 

1  5  Int.  for  1st  yr.  ""*'*"y'  ^^  ^^^  ^"^  *^^  ^"*^*'- 

est  of  $250  for  1  yr.  We  add 

$  2  6  5  Prin.  for  2d  jt.  t],i3  interest  to  the  principal, 

15.90  Int  for2cl  JT.  and  compute  the  interest  on 

$280.90  Prln.  for  3d  yr.  ^^'l^  amount  for  another  year. 

/I    o  1  T  »  i.     «  ^Vc  qdd  this  interest  to  the 

4.21  Int.  for  3  mo.  .     .     ,       ,    , 

prmcipal  as  before,  and  com- 

$285.11  Amount  for  2  yr.  8  mo.  p„te  interest  on  this  amount 

250.00  First  Principal.  for  3  months,  which  we  add 

$35,11  Comp.  Int.  for  2  j-r.  3  mo.        <"  ^^^^  principal.     From  this 

amount  we  subtract  the  orig- 
innl  i^rinoipal,  and  Obtain  $35.11,  the  compound  interest  required. 

IwLLi:. — Find  the  interest  of  tJie  principaJ  /^"'  fJ>^  ^>v'  ^vmmVv^ 
of  time  at  Vie  end  of  which  interest  is  due. 

Add  thii<  interest  to  the  principal,  and  compute  Uie  inienst 
ujwn  this  amount  for  Hie  next  period,  and  so  continue. 

Subtract  the  fpven  principal  from  the  last  amount ^  and  the  re- 
mainder will  be  the  compound  interest. 

1.  If  the  intt'rest  is  compounded  semi-annually,  the  rate  is  consid- 
ered as  one-half  the  annual  rate;  if  quarterly,  one-fourth,  etc. 
15  1  ..  . 


226  PERCENTAGE. 

2.  When  the  time  consbts  of  years,  montlis,  and  days,  find  the 
compound  interest  for  the  greatest  number  of  entire  periods,  and  to 
this  add  the  simple  interest  upon  the  amount  for  the  rest  of  the  time. 

2.  Compute  the  compound  interest  on  8315  for  2  yr.  6 
mo.  at  6^. 

3.  Find  the  amount  of  8324.18  f  '  .  5  mo.  at  7^ 
compound  interest. 

4.  What  is  the  compound  interest  on  8525.75  for  3  yr. 
4  mo.  at  6^  ? 

Computations  in  comix>und  interest  may  be  shortened  very 
much  by  the  use  of  the  table  on  the  following  page. 

5.  What  is  the  compound  interest  of  $325.10  for  3  yr. 
2  mo.  at  6^  ? 

Analysis. — By  referring  to  the  taUe,  the  amount  of  $1  for  3  yr.  ig 
found  to  be  $1.191016.  Computing  interest  on  this  sum  for  the  remain- 
ing 2  mo.,  the  amount  is  $1.202926.  Since  the  amount  of  $1  for  the 
given  time  is  $1.202926,  the  amount  for  8325.10  will  be  325.10  times 
that  sum.  If  from  this  product  the  principal  is  subtracted,  the  re- 
mainder is  the  compound  interest. 

6.  Find  the  compound  interest  of  8600.50  for  3  yr.  7  mo. 
at  6%. 

7.  Find  the  compound  interest  of  8318.25  for  2  yr.  4  mo. 
at  7^. 

8.  Find  the  compound  interest  of  8412.08  for  3  yr.  2  mo. 
10  da.  at  6^. 

9.  Find  the  compound  interest  of  8310.24  for  2  yr.  5  mo. 
15  da.  at  8^. 

10.  What  is  the  difference  between  the  simple  interest  on 
8328  for  2  yr.  7  mo.  at  7^,  and  the  compound  interest  on 
same  sum  for  the  same  time  at  6^^  ? 

11.  K  I  deposit  8300  in  a  savings  bank  which  compounds 
at  6^  semi-annually,  how  much  w411  be  due  me  in  3^  years  ? 


INTEREST. 


227 


COMPOUND    INTEREST   TABLE, 
Showing  the  amount  of  $1,  at  various  rates,  compound  int.  from  1  to  20  years. 


Yrs. 

2>^  per  cent. 
1.025000 

3  per  cent 

3^2  P^T  cent. 

4  per  cent 

5  per  cent. 

G  percent. 

1 

1.030000 

1.035000 

1.040000 

1.050000 

1.060000 

2 

1.050625 

1.060900 

1.071225 

1.081600 

1.102500 

1.123600 

3 

1.076891 

1.092727 

1.108718 

1.124864 

1.157625 

1.191016 

4 

1.103813 

1.125509 

1.147523 

1.169859 

1.215506 

1.262477 

5 

1.131408 

1.159274 

1.187686 

1.216653 

1.276282 

1.338226 

6 

1.159693 

1.194052 

1.229255 

1.265319 

1.340096 

1.418519 

7 

1.188686 

1.229874 

1.272279 

1.315932 

1.407100 

1.5036;}O 

8 

1.218403 

1.266770 

1.316809 

1.368569 

1.477455 

1.593848 

9 

1.248863 

1.304773 

1.362897 

1.423312 

1.551328 

1.689479 

10 

1.280085 

1.343916 

1.410599 

1.480244 

1.628895 

1.790848 

11 

1.312087 

1.384234 

1.459970 

1.539454 

1.710339 

1.898299 

12 

1.344889 

1.425761 

1.511069 

1.601082  1  1.795856 

2.012197 

13 

1.378511 

1.468534 

1.563956 

1.665074  !  1.885649 

2.132928 

14 

1.412974 

1.512590 

1.618695 

1.731676 

1.979932 

2.260<>04 

16 

1.448298 

1.5-57967 

1.675349 

1.800944 

2.078928 

2.396558 

16 

1.484506 

1.604706 

1.73.3986 

1.872981 

2.182875 

2.540352 

17 

1.521618 

1.6.V2848 

1.794676 

1.947901 

2.292018 

2.692773 

18 

1.559659 

1.702433 

1.857489 

2.025817 

2.406619 

2.854339 

19 

1.598650 

1.753506 

1.922501 

2.106849 

2.5269r)0 

3.025600 

20 

1.638616 

1.806111 

1.989789 

2.191123 

2.653298 

3.207136 

Yis. 

7  per  cent. 

Spcrceiir. 

9  per  cent 

10  percent 

11  per  cent 

12  per  cent 

1 

1.070000 

1.080000 

1.090000 

1.100000 

1.110000 

1.120000 

2 

1.144900 

1.166400 

1.188100 

1.210000 

1.232100 

1. 251-100 

3 

1.225043 

1.269712 

1.295029 

1.331000 

1.367631 

1.401«i08 

4 

1.310796 

1.360489 

1.411582 

1.464100 

1.518070 

l.57:r.l9 

5 

1.402552 

1.469328 

1.538624 

1.610510 

1.685058 

1.762342 

6 

1.500730 

1.586874 

1.677100 

1.771561 

1.870414 

1.973822 

7 

1.005781 

1.713824 

1.828039 

1.948717 

2.076160 

2.210681 

8 

1.718186 

1.850930 

1.992563 

2.143589 

2.304537 

2.47-.9(i3 

9 

1.838459 

1.999005 

2.171893 

2.357948 

2.558036 

2.773078 

10 

1.967151 

2.15892.5 

2.367364 

2.593742 

2.839420 

3.105sts 

11 

2.10^1^52 

2.3r^l  61^0 

2.580126 

2.853117 

3.151757 

3.47S-,  10 

12 

■•  '  ■ 

3.138428 

3.498450 

s.y 

13 

3.452271 

3.883279 

4.:;' 

14 

^.•»<  r>.).,  i  j..:t,\,  r.n 

....>  n  ,zi 

3.797498 

4.310440 

4.ss,iii 

15 

2.7bmn   1  3.172169 

3.642482 

4.177248 

4.784588 

5.47;  ;•-.(;•■> 

16 

')  <i-.-M.:  1   ■>  |.)-.M  I" 

•.•  ..-,i-,>,- 

!  -.(1  lu-'J 

-.  ..,,,.,».) 

6.i;i0392 

17 

! 

6.866040 

18 

■i 

7.689964 

19 

3.016527  4.315701 

5.14161)1 

6.1 15909 

7.263342 

8.612760 

20 

3.869684  4.660957 

5.604411 

6.727500 

8.062309 

9.646291 

228  PERCENTAGE. 


ANNUAL  INTEREST. 

339.  Annual  Interest  is  simple  interest  on  the  prin- 
cipal and  upon  any  interest  overdue,  when  the  contract  con- 
tains the  words,  "with  annual  interest,"  or,  "with  interest 
payable  annually." 

Annual  interest  is  not  considered  l^al  in  some  States. 

1.  Find  the  amount  of  $3500  for  4  yr.  6  mo.,  with  inter- 
est payable  annually  at  6j^. 

PROCESS.  Analysis. — Since 

Int.  of  $3500  for  4^  yr.  =   $945  »"""^^  mUiTeat  is  sim- 

Int.  of  $210    for  8    yr.  =   $100.80       P^?    1"^"''^  °"   ^^^^ 

•^  principal    and     upon 

Annual  Interest      =  $1045.80        any  over-due  interest, 

$3500 +  $1045.80 --$4545.80.  Amt.      ^^'^  ^^^  ^"^  '^^  >?^^^- 

est  upon  the  principal, 
which  is  $945,  and  then  upon  the  interest  due.  The  interest  for  each 
year  is  $210.  The  interest  for  the  first  year  remained  unpaid  for  3J 
years;  that  for  the  second  year,  2\  years;  that  for  the  third  year,  1^ 
years;  and  that  for  the  fourth  year,  for  ^  year;  therefore  the  annual 
interest,  $210,  drew  interest  for  S^-fS^  +  H+i  years,  or,  8  years, 
which  is  $100.80.  This  sum,  added  to  the  simple  interest,  $945  = 
$1045.80,  the  annual  interest.  Tliis  sum  added  to  the  principal, 
$3500  =  $4545.80,  the  amount  due. 

Rule. — Compute  the  interest  on  the  principal  for  the  entire 
time,  and  on  each  year's  interest  from  the  time  it  was  due  up  to 
Vie  end  of  the  period. 

The  sum  of  these  interests  vM  he  the  annual  interest. 

2.  How  much  is  due  upon  a  note  of  $350  which  has  run 
4  years,  interest  at  8^,  payable  annually? 

3.  How  much  was  due  April  15,  1877,  on  a  note  for  $750, 
dated  Jan.  1,  1873,  with  interest  at  6^,  payable  annually? 


INTEREST.  229 


PARTIAL    PAYMENTS. 

340.  A  Partial  Payment  is  a  payment  in  part  of  a 
note  or  other  obligation. 

341.  An  Indorsement  is  the  statement  of  the  amount 
of  a  payment  and  the  time  when  it  was  made.  It  is  written 
on  the  back  of  the  note  or  other  written  obligation. 

342.  Business  men  often  settle  notes  and  accounts  running 
for  one  year  or  less  by  what  is  known  as  the  Mercantile 
Itiilr. 

Mercantile  Rule. — Find  tlie  amount  of  the  principal  at  the 
thrte  of  settlement. 

Find  iJie  amount  of  ea^h  payment  from  the  time  it  was  made 
until  the  time  of  settlement,  and  from  the  amount  of  (lie  principal 
subtract  tlie  amounts  of  the  payments. 

1.  A  note  for  $850,  on  demand  with  interest  at  7j^  dated 
Jan.  1,  1876,  was  indorsed  as  follows:  April  10,  1876,  §200; 
Sept.  15,  1876,  3255.     How  much  was  due  Nov.  15,  1876? 

2.  What  is  the  balance  due  at  the  end  of  a  year  on  a  note 
for  81800,  dated  May  15,  1875,  on  which  the  following  pay- 
ments had  been  made:  Sept.  20,  1875,  $300;  Jan.  18,  1876, 
S200;  AprU  20,  1876,  $1000;  when  the  rate  is  7^  ? 

3.  $585.25.  CiiAULESTON,  S.  C,  March  3, 1876. 
Eight  months'  after  date,  for  value  received  I  premise  to  pay  to 

the  order  of  E.  S.  Farran,  Five  Hundred  Eighty^ve  -^  Dollars, 
with  hUerea  at  Ifc-  H.  S.  Ladphier. 

This  note  was  indorsed  as  follows:  June  8,  1876,  $325; 
Aug.  4,  1876,  $84.30;  Sept.  2,  1876,  $100.  What  was  due 
on  the  note  at  maturity? 


230  PERCENTAGE. 

34»].  Most  of  the  States  have  adopted  the  United  States 

Itule  for  computing  the  amount  due  ujwn  any  obligation 
where  partial  payments  nro  ni;i<lo,  based  upon  tin.  follnuinn^ 
principle. 

Principle. — The  indebtedness  thmdd  he  computed  wfienever  a 
jxnjment  is  madej  hut  Uie  principal  must  not  he  increased  hy  the 
addition  of  interest. 

WRJ  I   I  I    \     EXERCISES, 

1.  A  note  was  given,  Jan.  1,  1870,  for  $700.  The  fol- 
lowing payments  were  indorsed  ujwn  it:  May  6,  1870,  $85; 
July  1, 1871,  $40;  Aug.  20, 1871,  $100;  Jan.  10, 1873,  $350. 
How  much  was  due  Sept.  30,  1874,  interest  at  6%  ? 

PROCESS. 

Principal $700.00 

Int.  to  May  6,  1870,-4  mo.  5  da. _1^-^? 

Amount 714.58 

First  payment 85.00 

New  Principal G29.58 

Int.  from  May  6,  1870,  to  July  1,  1871—1  yr.  1  mo. 

25  da 43.55 

Second  payment,  less  than  interest  due  .       $40.00 

Int.  on  $629.58  from  July  1,  1871,  to  Aug.  20, 1871,— 

1  mo.  ly  da 5.14 

Amount 678.27 

Third  payment  to  be  added  to  second  .        .        .          $100  J  40.00 

New  Principal 538.27 

Int.  from  Aug.  20, 1871,  to  Jan.  10, 1873,-1  yr.  4  mo. 

20da.      .        .        .                 44.85 

Amount 583.12 

Fourth  payment 350.00 

New  Principal 233.12 

Int.  from  Jan.  10,  1873,  to  Sept.  30, 1874,-1  yr.  8  mo.  20  da.  24.08 

Amount  due,  Sept.  30,  1874     .        .                .  $257.20 


PARTIAL  PAYMENTS.  231 

United  States  Rule. — Find  Vw  amount  of  the  jmndpal 
tc  a  time  when  a  payment^  or  Hie  sum  of  Hie  'payments^  equals 
or  exceeds  the  intered  due,  and  from  Hm  amount  subtract  such 
payment  or  payments.  WitJi  the  remainder  as  a  new  principaly 
proceed  as  before, 

2.  A  note  for  $2500,  dated  July  10,  1871,  bore  the  fol- 
lowing indorsements:  Sept.  15,  1871,  S150;  Nov.  12,  1871, 
$300;   Dec.  1,  1871,  $100;    AprU  3,  1872,  $325;    May  15, 

1872,  $275;  Sept.  20,  1872,  $1000.  How  much  was  due 
Jan.  1,  1873,  the  rate  of  interest  being  6%? 

3.  How  much  was  due  at  maturity  on  a  note  for  $2150, 
dated  Sept  20,  1873,  to  run  2  years  6  months,  on  which  the 
following  payments  were  indorsed:  Dec.  15,  1873,  $75;  Feb. 
4;  1874,  $200;  April  3,  1874,  $150;  July  1,  1874,  $500; 
Dec.  16,  1874,  $1000,  the  rate  of  interest  being  8%? 

4.  A  note  for  $6725,  dated  Feb.  10, 1875,  had  the  following 
indorsements:  May  5,  1875,  $275;  Aug.  15,  1875,  $50;  Nov. 
12,  1875,  $1000;  Jan.  3,  1876,  $184.25;  Sept.  13,  1876, 
$84.10;  Dec.  23,  1876,  $1000.  How  much  was  due  Fob.  10, 
1877,  interest  at  6^  ? 

5.  A  bond  was  given  April  4,  1870,  for  $5825,  wiili  inter- 
est at  8%.  The  following  payments  were  indorsed  upon  it: 
May  15, 1871,  $728.50;  April  8,  1872,  $1000;  Dee.  12, 1872, 
$125;  July  9,  1873,  $980;  June  12,  1874,  $1000.  How 
much  remained  due  upon  the  bond  April  4,  1875? 

6.  Sept.  25,  1872,  James  Hanna  gave  his  note  for  $895.75 
with  interest  at  10%.     He  paid  on  it  as  follows:   Jan.  10, 

1873,  $25;  Oct.  12,  1873,  $200;  Jan.  18,  1874,  $75;  March 
25,  1874,  $187.50;  Jan.  1,  1875,  $375.  Hmv  much  was  due 
when  he  paid  the  note,  Nov.  15,  1875? 

7.  Kequired,  the  balance  due  on  a  note  dated  Jan.  1,  1875, 
for  $580  at  5^,  to  run  2  years,  "»•  wKwh  a  j^ivmcnt  of  $85 
was  made  every  3  months. 


232  PERCENTAGE. 

8.  A  note  for  810000,  with  interest,  dated  Milwaukee,  Wis., 
Dec.  12,  1875,  was  indorsed  as  follows:  Feb.  23,  1876,  $750; 
July  17,  1876, 8108.25;  Nov.  23, 1876,  83000;  Jan.  18, 1877, 
84000.    How  much  was  due  May  12,  1877,  interest  at  Sfc  ? 

9.  Required,  the  Iwlance  due  July  8,  1876,  on  a  note  for 
63124.75,  at  S^  interest,  dated  Feb.  15,  1874,  on  which  a 
l)ayment  was  made  Dec.  23, 1874,  of  8985;  another  of  8875.35, 
Feb.  15,  1875;  another  of  81025,  Feb.  20,  1876. 

10.  $1885.75.  Detuoit,  Mich.,  Feb.  10,  1874. 

Tvx>  yean  after  dale,  for  value  received  I  promise  to  pay 
W.  W.  Heilbromier  or  order,  One  Thousand  Eigld  Hundred 
Eighty-five  ^  Dollars,  icith  interesL  q  jj  VroRARD. 

On  this  note  were  the  following  indorsements:  June  30, 

1874,  850;  Nov.  8, 1874,  8100;  Feb.  5,  1875,  8125;  April  17, 

1875,  8500;  Dec.  1,  1875,  8500.     How  much  remained  un- 
piid  Mar.  1,  1876? 


PROBLEMS  IN  INTEREST. 

PROBLEM  I. 

344.  Principal,   rate,   and   time  given,  to  find   the 
interest. 

This  lias  already  been  solved.     (See  page  220.) 

Rule. — Multiply  Uie  interest  of  81,  for  the  given  rate  aiid 
time,  by  the  principal. 

PROBLEM  IT. 

345.  Principal,  rate,  and  interest  given,  to  find  the 
time. 

1.  How  much  is  the  interest  of  §100  for  a  year  at  6^  ? 
For  2  years?     For  3  years? 


PARTIAL   PAYMENTS.  16d 

2.  When  8100,  loaned  at  G^,  brings  an  income  of  812, 
for  how  long  a  time  was  it  loaned  ?  How  long  when  the  in^ 
terestwas318?    824?    83?    84?  .  82?    81.50? 

3.  When  850,  loaned  at  10^,  brings  an  income  of  810, 
for  how  long  a  time  was  it  loaned? 

RxjLE. — Divide  the  given  interest  by  the  interest  of  the  prin- 
cipal for  one  year. 

In  what  time  will 

4.  8250  produce  830  interest  at  6^  ? 

5.  8600  produce  824  interest  at  8^  ? 

6.  8115  produce  813.80  interest  at  6^  ? 

7.  812.60  produce  84.15  interest  at  7%  ? 

»  8.  835.25  produce  8^3.25  interest  at  7^  ? 

9.  825  produce  825  interest  at  6^  ? 

10.  8150  double  itself  at  Sfo  ? 

11.  Any  sum  double  itself  at  5^  ?     6^  ?     7%  ? 

12.  Any  sum  triple  itself  at  5^  ?     6^  ?     7^  ? 


PROBLEM  IIL 

346.  Prineiiml,  time,  and  interest  given,  to  And  the 
rate. 

1.  What  is  iho  miorest  of  8100  for  1   year  at  1^?     At 
2^?    At  3;;, 

2.  When  the  mtercst  of  8100  for  1  year  was  §8,  what  was 
the  rate? 

3.  When  the  interest  of  8100  for  2  years  was  814,  what 
was  the  rate? 

4.  When  the  interest  of  850  for  3  years  was  $15,  what 
was  the  rate? 

Rule. — Divide  the  given  iuteresty  by  the  interest  of  the  prin- 
cipal for  the  given  time^  at  1  i>cr  cent. 


234  PERCENTAGE. 

What  is  the  rate  i)er  cent  when  the  interest 

5.  Of  $125  for  2  years  is  $15? 

6.  Of  $250  for  6  months  is  $8.75? 

7.  Of  $415  for  2  years  G  months  is  $56,025? 

8.  Of  $317  for  1  year  5  months  is  $31.44? 

9.  Of  $215  for  2  years  7  months  10  days  is  $39.30? 

10.  Of  $325.18  for  5  months  26  days  is  $11.13? 

11.  Of  $30.18  for  2  months  3  days  is  $  .32? 

12.  Of  $24.36  for  3  months  3  days  is  $  .44? 

13.  Of  $25,40  for  1  mouth  15  days  is  $  .397? 


PROBLEM  IV. 

347.  Rate,  time,  and  interest  given,  to  find  the  prin- 
eipal. 

1.  At  6^,  what  sum  will  produce  $6  yearly? 

2.  At  6^,  what  sum  will  produce  $12  yearly?  What 
$18  yearly?  What  $12  in  two  years?  What  $18  in  2 
years? 

Rule. — Divide  the  given  interest  by  the  interest  of  $1,  for  the 
given  time  at  Uie  given  rate. 

What  sura  of  money  will  produce 

3.  $36.60  interest  in  2  years  at  6^  ? 

4.  $35.70  interest  in  2  years  6  months  at  8^  ? 

5.  $51.20  interest  in  5  years  6  months  at  5^  ? 

6.  $50.84  interest  in  6  months  27  days  at  6^  ? 

7.  $39.18  interest  in  5  months  18  days  at  6^  ? 

8.  $41.25  interest  in  3  months  15  days  at  9^  ? 

9.  $87.50  interest  in  1  month    12  days  at  7fc  ? 

10.  $68.75  interest  in  3  months  10  days  at  6^  ? 

11.  $50.83  interest  in  2  years  3  months  at  Qfc  ? 

12.  $81.25  interest  m  3  years  5  months  at  8^  ? 


I'EiiCKMAGE.  235 


NOTES. 

348.  A  Note,  or  JProinissory  Note,  is  a  written 
promise  to  pay  a  sum  of  money  at  a  given  time. 

.*U9.  The  Maker  or  Draiver  is  the  person  who  signs 
the  note. 

350.  The  JPdi/ee  is  the  jierson  to  whom  it  is  made 
payable. 

351.  The  Holder  is  the  person  who  has  the  note  in 
his  possession. 

352.  The  Indorser  is  the  person  who  writes  his  name 
npon  the  back  of  the  note  to  transfer  it  or  guarantee  its 
payment. 

The  payee  may  indorse  by  writing  his  name  on  the  back  of  a  note. 
It  18  then  payable  to  the  hearer.     He  may  alSo  indorse  by  writing, 

"  Pay  to  A B ,  or  order."     It  then  requires  the  signature  of 

A B before  it  is  payable. 

353.  The  IPace  of  a  note  is  the  sum  named  in  it. 

354.  A  Nef/ot table  Note  is  a  note  payable  to  the 
order  of  the  payee  or  bearer. 

355.  A  No u -negotiable  Note  is  a  note  payable  to 
the  2>a!/ee  only. 

Notes  sliould  contain:  the  date,  the  time  when  due,  the  amount  of 
the  note  written  in  words,  the  words  "for  value  received,"  and  "with 
interest,"  if  such  is  the  contract. 

356.  Thrve  Days  of  Grace  are  usually  allowed  after 
the  specified  time,  before  the  note  is  said  to  mature  or  he  due, 
except  on  notes  j)ayable  on  demand,  and  where  the  words 
"without  grace"  are  written. 


236  PERCENTAGE. 

If,  when  a  note  is  unpaid  at  its  maturity,  the  holder  fails 
to  notify  the  indorsers  of  the  fact,  they  are  released  from 
rcsix)nsibility  regarding  its  payment. 

357.  Notes  without  interest  draw  interest  at  the  legal  rate 
after  they  become  due,  but  a  note  does  not  draw  interest 
until  after  it  is  due,  unless  the  words  **with  interest/*  or 
"with  use,**  arc  written  in  it 

FORMS  OF  NEGOTIABLE  NOTES. 

$729.18.  Cincinnati,  O.,  Oct.  5,  1876. 

For  value  received^  two  months  after  date  I  promise  to  pay 
James  J.  Coney  or  order,  Seven  Hundi-ed  Twenty-Nine  -^  Dd- 
lars,  witii  biterest,  jj    jj    Blckham. 

$600,  DETRorr,  Jan.  29,  1877. 

For  v(dne  received^  three  monUis  after  date  I  promise  to  pay 
11.  G.  Bvrlingame/or  hearei',  Six  JT'-^'^h-rff  Dollars. 

W.  H.  Sargent. 

WBITTBJr    EXEnCISES. 

1.  'W'rite  a  negotiable  note  for  §500.25,  making  yourself 
the  payee,  and  James  J.  Rogers  the  maker.  Interest  at  the 
legal  rate. 

2.  Write  a  non-negotiable  note  for  8315.17,  making  W.  R. 
Howard  the  payee,  payable  on  demand  without  interest. 

3.  Write  two  forms  of  negotiable  notes  for  83184.25,  due 
in  three  months  to  James  P.  Hermann,  with  interest. 

4.  Indorse  them  properly  for  transferring;  one  to  bearer, 
and  the  other  to  H.  H.  Hurd,  or  order. 

5.  Write  a  note  from  the  following  data:  Face,  85000; 
negotiable;  maker,  .P.  G.  Sloane;  payee,  J.  S.  Orton;  pay- 
able on  demand;   rate  of  interest,  the  legal  rate. 


DISCOUNT.  237 


COMMERCIAL  DISCOUNT. 

858.  Discount  is  an  allowance,  or  deduction,  made 
from  ii  sum  of  money  to  be  paid. 

Problems  in  discount  may,  or  may  not,  have  reference  to 
time. 

359..  Conunercial  Discount  is  a  deduction  from 
the  jprtce  of  an  article,  or  from  a  hlUy  without  regard  to  time. 

360.  The  Net  Price  is  the  selling  price,  minus  the 
discount. 

361.  The  Cash  Value  of  a  bill,  is  its  face  less  the  dis- 
count. 

EXEBCISES. 

362.  1.  If  gloves  marked  at  $1.50  per  pair,  are  sold  at 
10%  discount,  what  is  the  discount?     AVhat  is  the  net  i)rice? 

2.  If  flour  is  offered  at  S7.50  per  barrel,  with  a  discount 
of  5%  for  cash,  what  is  the  discount? 

3.  Find  the  valuo  of  a  bill  of  goods,  amounting  to  $845 
at  5^  discount. 

4.  Find  the  value  of  a  bill  of  goods  amounting  to  $680, 
when  a  discount  of  2\%  is  allowed  for  cash. 

5.  What  is  the  cash  value  of  a  bill  of  goods,  amounting 
to  $3215.45  at  20^  discount,  and  5^  off  for  cash? 

In  problems  like  No.  5,  it  is  always  understootl  that  the  per  cent. 
off  for  casih  is  to  be  reckoned  upon  the  price  after^the  previous  discounts 
have  been  allowed. 

6.  What  is  the  cash  value  of  a  bill  amountincr  to  $3750 
at  10%  discount,  and  2\fc  off  for  cash? 

7.  What  is  the  cash  value  of  a  bill  of  good^  amounting  to 
$2157.25  at  15%  discount,  and  3%  off  for  cash? 


238  PERCENTAGE. 


TRUE  DISCOUNT. 

303.  1.  What  will  be  the  amount  of  $100  in  1  year,  in- 
terest lit  6^  ?     In  2  years?     In  3  years?     In  4  years? 

2.  What  is  the  value  now  of  8106,  to  be  paid  in  1  yr., 
when  money  loans  at  6^  ?    Of  1112,  to  be  paid  in  2  yr.  ? 

3.  AVhat  is  the  value  now  of  $212,  to  be  paid  in  1  yr., 
money  loaning  at  6^?     Of  $224,  to   be   paid  in  2  yr.? 

4.  What  is  the  worth  now  of  a  debt  of  $535,  to  be  paid 
in  1  yr.,  when  money  can  be  loaned  at  7^^? 

f).  What  is  the  present  value  of  8672,  due  in  IJ  yr.,  when 
money  is  loaning  at  8^?  Of  $316,  due  in  2  yr.,  money 
being  worth  8^? 

364.  True  jyiscount  is  a  deduction  made  for  the  pay- 
ment of  a  sum  of  money  before  it  is  due. 

305.  The  JPresent  Worth  of  a  sum  of  money  due  at 
some  future  time,  is  a  sum  which,  put  at  interest  at  the 
specified  rate,  will  amount  to  the  debt  when  it  becomes  due. 

WRITTEN    EXERCISES, 

3G6.  1.  What  sum  of  ready  money  is  equivalent  to 
8784.25,  payable  in  2  years,  when  money  is  worth  7^  ?  ' 

paocEss.  Analysis. — Since 

$1.14  =  Amount  of  $1  for  2  yr.  ^^"^  ''°"'"-  "  P"'  ^' 

^  ''  interest    now    at    7^ 

$784.25-^1.14  =  8687.94.  ^^^^A  amountto$l.l4 

8687.94  =  Present  Worth.  in  2  years,  it  Avill  re- 

$784.25—8687.94  =  896.31  Discount.  q»ire  as  many  dollars 

now,  to  amount  to 
$784.25  in  2  years,  as  $1.14  is  contained  times  in  $784.25,  which  is 
687.94  times.     Therefore,  the  present  worth  is  $687.94. 

The  face  of  the  debt,  $784.25  —  8687.94,  leaves  $96.31,  the  discount. 


DISCOUNT.  239 

Rule. — Divide  the  amount  due  by  the  amount  of  $1,  for  Hie 
given  tune  and  rate,  and  Vie  quotient  wHl  he  Hie  present  wortJi. 

Subtract  the  present  woHh  from  the  amount  due,  and  tJie  re- 
muiiuler  will  be  Uie  true  discowd. 

What  are  the  present  worth  and  discount  of  the  following: 

2.  §975.50,  payable  in  1^  years,  when  money  is  worth  6^  ? 

3.  $845.20,  payable  in  1 J  years,  when  money  is  worth  7^  ? 

4.  $958.75,  payable  in  3    years,  when  money  is  worth  7^  ? 

5.  $576.25,  payable  in  3    years,  when  money  is  worth  8^  ? 

6.  $8575,  payable  in  2  yr.  3  mo.  10  da.,  when  money  is 
worth  5J^  ? 

7.  $4274,  payable  in  1  yr.  4  mo.  15  da.,  when  money  is 
worth  Qff)  ? 

.  8.  $2845,  payable  in  3  yr.  6  mo.   15  da.,  when  money  is 
worth  7fc  ? 

9.  $1752.75,  payable  in  1  yr.  3  mo.  20  da.,  when  money  is 
worth  8^  ? 

10.  $5493.50,  payable  in  2  yr.  5  mo.  25  da.,  when  money  is 
worth  9<^? 

11.  $3457.84,  payable  in  7  mo.  10  da.,  when  money  is 
worth  Hfc  ? 

12.  Bought  a  ferm  for  $14500  cash,  and  sold  it  for 
$16000,  payable  one-half  cash  and  the  remainder  in  one 
year.  To  what  amount  of  cash  in  hand  was  the  selling  price 
equal  when  money  was  loaning  at  6j^  ? 

13.  A  merchant  bought  a  bill  of  goods  amounting  to  $5275, 
on  3  mo.  credit,  but  was  offered  2^^  discount  for  cash.  How 
much  would  he  gjiin  by  l>aying  cash,  money  being  worth  S%  ? 

14.  A  mercliant  holds  two  notes,  one  for  $187.25,  due 
Feb.  15th,  1877,  and  the  other  for  $382.75,  due  April  Ist, 
1877.  Wliat  would  be  due  him  in  cash,  on  both  notes, 
Jan.  Ist,  1877,  money  being  worth  8^  ? 


240  PERCENT  AOE. 

1  ">.  If  I  pay  a  debt  of  $4725  1  yr.  3  mo.  !:>  .la.  before  it 
is  due,  what  discount  should  be  made  from  tl»e  face  of  the 
debt,  money  being  worth  8^? 

16.  Bought  850  barrels  of  pork  at  $21.50  per  barrel,  on 
3  mo.  credit,  and  sold  it  the  same  day  for  621.50  per  barrel, 
cash.     How  much  did  I  gain,  money  being  worth  (y^%  ? 

17.  How  much  will  I  gain  if,  instead  of  laying  65400, 
cash,  for  a  piece  of  property,  I  pay  60000  in  1  yr.  J  mo., 
money  being  worth  9^  ? 

18.  How  much  should  be  discounted  on  a  bill  of  63725.87, 
due  in  8  mo.  10  da.,  if  it  is  paid  immediately,  money  being 
worth  lOj;^  ? 

19.  Bought  250  barrels  of  granulated  sugar,  weighing  282 
lb.  each,  at  10 J  cents  per  lb.,  on  1  mo.  credit.  How  much 
cash  would  settle  the  account,  money  being  worth  8%? 

20.  A  crockery  dealer  bought  63725  worth  of  goods,  for 
cash,  at  a  discount  of  374^  from  the  regular  list  prices,  and 
sold  them  at  the  regular  list  prices  on  4  months'  time.  ILjw 
much  did  he  gain,  money  being  worth  S%  ? 


BANK  DISCOUNT. 

367.  A  J^anL'  is  an  institution  established  by  law  to 
receive  money  for  safe-keeping,  to  loan  money,  or  to  issue 
notes  or  bills  to  circulate  as  money. 

368.  A  Chech  is  a  written  order  upon  a  bank  for  money 
deposited  with  it. 

369.  Sank  Discount  is  the  simple  interest  on  the 
amount  due,  paid  in  advance,  for  3  days  more  than  the 
specified  time. 

370.  The  Proceeds^  or  Avails,  of  a  note  are  the 
face  of  the  note,  less  the  discount. 


BANK    DISCOUNT.  241 

371.  The  Maturity  of  a  note  occurs  on  the  last  day 
of  grace. 

If  the  la«t  day  of  grace  falls  on  Sunday  or  a  legal  holiday  the  note 
matures  one  day  earlier. 

372.  The  Tertn  of  Diseotint  is  tlic  numbor  of  days 
from  the  time  of  discounting  to  the  maturity  of  the  note. 

1.  Banks  usually  discount  for  short  terms,  not  exceeding  3  mo.  or 

4  months. 

2.  Unless  otherwise  specified,  30  days  are  a  month ;  65  day.s,  2  mo. 

5  da.,  etc. 

3.  Bank  discount  is  undoubtedly  wrong  in  principle,  but  has  been 
sanctioned  by  custom. 

CASE  I. 

373.  To  find  (lie  Bank  Discount  an<l  Procee<l8. 

1.   Wh:it  is  the  bank  discount  of  8385  for  3  mo.  at  6^? 

PROCESS.  Analysis.  —  Since 

$385,  Face  of  Note.  ^^"^^^  '"  discounting, 

.0155,  Rate  of  Discount.  ''^^"'^^  the  simple  in- 

terest  in  advance,  we 


$5.9675,  Discount.  find    the    interest    of 

$385— $5.9675  =  $379.0325,  Proceeds.     ^^^  ^«»-  ^  mo.  3da.  at 

6fc,  which  is  $5.9675. 
And  since  the  present  worth  is  the  face  of  the  debt  minus  the  dis- 
I  ount,  we  subtfar*  ""  " '"''  from  $385,  which  gives  $379.0325,  the  pro- 
'  y-isla  or  avails. 

Rule. — To  find  die  bank  discount:  Compute  Hie  interest  on 
the  faee  of  the  note  for  3  days  more  Vian  Hie  specific  time  at  the 
(jiuen  rate. 

To  find  (lie  proceeds:  Subtract  die  bank  discount  from  Vie  fare 
of  the  note. 

If  the  note  bears  interest,  find  the  discount  on  the  amount  of  th« 
note  at  its  maturity.  This  itmouiU,  less  the  diflCGunt,  Will  be  the  pro- 
ceeds. 


242  PERCENTAGE. 

Find  the  bank  discount  of: 

2.  $  318.25  for  2  mo.  15  da.  at  6j?. 

3.  83846.18  for  1  mo.  17  da.  at  Ifc 

4.  82184.39  for  3  mo.  10  da.  at  7;^. 

5.  $  824.17  for  3  mo.  28  da.  at  8^. 

6.  84484.15  for  3  mo.  18  da.  at  7^. 

7.  87318.69  for  2  mo.  15  da.  at  6^. 

8.  88984.15  for  30  da.  at  6^. 

9.  8  765.30  for  65  da.  at  6^. 

10.  $S75.  Chicago,  III.,  Dec.  20, 187C 
Tioo  vwnths  c^Ur  date^  for  value  reeeiv&l  I  promise  to  jxiy  E. 

D.  Brotison,  or  order^  Three  Humlred  SevetUy-Five  Dollars^ 
with  interest  at  10^ ,  at  tiie  I^rst  National  Bank,  ChieagOf  III, 

S.  Howard  Blackwell. 

This  note  was  discounted  Jan.  23,  1877,  at  lO^J^.  What 
Avas  the  Iwink  discount?     Whiit  wore  tl>'^  |'?'"eeds? 

11.  What  is  the  luiiik  iliscount  on  a  noii-  for  8890.2')  at 
2  months,  the  rate  being  7%  ?    What  are  the  proceeds? 

12.  What  is  the  difference  between  the  btmk  dLscount 
and  the  true  discount,  on  a  note  for  3  months  for  83725.85, 
the  rate  being  8$^? 

13.  $15,725.95.  St.  Paul,  Mujn.,  May  15,  1876. 
For  value  received ,  tioo  months  after  date  **  The  North  River 

Sugar  Refining  Co,** promises  to  pay  Messrs.  Smith  ct*  Haughton, 
or  order.  Fifteen  Thousand  Seven  Hundred  Tiventy-five  ■^\ 
Dollars,  at  Tlw  Merchants  National  Bank,  New  Orleans,  La. 

North  Rfv^er  Sugar  Refining  Co., 
By  C.  Moller,  Sec^y. 

The  above  note  was  discounted  May  25,  1876,  at  6%, 
What  was  the  discount?     What  were  the  proceeds? 


BANK    DISCOUNT.  243 

14.  Mr.  A.  sold  goods  amounting  to  $3782.75,  and  received 
in  payment  a  note  at  3  mo.,  without  interest,  whicli  he  had 
discounted  at  a  bank  after  holding  it  1  mo.  How  much  did 
he  realize  from  the  sale,  the  rate  of  discount  being  8%? 

15.  A  merchant  sold  32  yards  of  silk  velvet  at  $8.75 
per  yard ;  48  yards  of  silk  at  82.80  per  yard ;  25  yards  of 
English  broadcloth  at  $6.25  jKjryard;  receiving  in  payment 
a  bank  note  payable  in  1  mo.  15  da.,  without  interest,  which 
he  had  discounted  on  the  same  day  at  9%.  How  much  did 
he  realize  in  cash  from  the  sale? 


CASE  II. 

374.  To  find  tlie  face  or  a  note  wheu  the  proc€K^«ls, 
time,  and  rate  are  given. 

'1.  What  is  the  bank  discount  of  $1  for  2  mo.  27  da.  at 
6%  ?     What  the  proceeds? 

2.  Since  8  .985  is  the  proceeds  of  $1  when  discounted  at 
2  mo.  27  da.  at  6%,  of  what  sum  is  $1.97  the  proceeds  for 
the  same  time  and  nite?  Of  what  sum  is  82.955  tho  pro- 
ceeds for  the  same  time  and  rate? 

3.  Of  what  sum  is  $394  the  j>roceeds,  when  discounted  at 
a  bank  for  1  mo.  27  da.  at  6%?  Of  \vli:if  .^iin  is  ?U)7  the 
proceeds 

4.  Of  \vluit  sum  i:i  8396  the  proceeds,  when  discounted  at 
a  bank  for  1  mo.  27  da.  at  6%? 

5.  The  proceeds  of  a  note  discounted  at  a  bank  for  2  mo. 
12  dM.  :it  7^/  "TO  $1182.50.     What  was  tho  face  of  it? 

PROCESS.  Analysis. — 

$1— $.0145J  =  $.9854i  Thoprocccdsof  $1.        Since  the  pm- 

S1182.50-4-.98544  =  $1200        The  face  of  Uie  note.      '^'''''^1  "L  f/ 
•  arc     $.OS.>l.^, 

$1182.50  are  the  proceedg  of  88  many  dollars  as  $.9854^  is  contained 

times  in  $1182.50,  which  is  1200  times.    Therefore  the  face  of  the  note 

was  $1200. 


244  PERCENTAGE. 

Rule. — Divide  the  proceeds  by  the  proceeds  of$l^  at  the  given 
rate  for  3  days  more  dian  the  specified  time. 

6.  The  proceeds  of  a  note  discounted  at  a  bank  for  2  mo. 
at  6%  &re  8989.50.     What  was  its  face? 

7.  What  must  be  the  f;ue  of  a  note  so  that  when  dis- 
counted at  a  bank  for  3  iii->  at  6^,  the  proceeds  will  be 
«1969? 

8.  A  merchant  wishes  to  ux'  S975,  which  he  can  secure 
by  giving  a  bank  note  payable  in  2  mo  ;  i  discounted  at 
7%.     For  what  sum  must  the  note  be  written '! 

9.  'For  what  sum  must  a  note  be  drawn  payable  in  3  mo., 
80  that  when  discounted  at  6%  the  proceeds  may  be  $1000? 

10.  A  man  owes  a  debt  of  $1375.38,  which  he  can  meet 
by  giving  his  note  payable  in  3  mo.,  discounted  at  9^.  For 
what  sum  must  the  note  be  written  so  that  the  avails  may 
be  just  large  enough  to  discharge  the  debt? 

11.  For  what  sum  must  a  note  be  drawn,  payable  in  15 
davs,  so  that  when  discounted  at  8^  the  proceeds  may  be 
$1257.25? 

12.  A  speculator  wishes  to  raise  $5250  on  an  indorsed 
note,  payable  in  1  mo.  15  da.,  discounted  at  7%.  For  what 
amount  must  he  draw  the  note  so  that  the  proceeds  may  be 
exactly  that  sum? 

13.  For  what  amount  must  a  note  be  made  payable  in 
3  mo.,  and  discounted  at  12^%,  so  that  the  proceeds  may  be 
§1875? 

14.  A  merchant  wishes  to  raise  6500  at  a  bank  by  a  note 
for  2  mo.  For  what  sum  must  the  note  be  drawn  that  he 
may  receive  $500  in  cash  from  the  banker  after  paying  the 
discount  at  8%? 

15.  For  how  large  a  sum  must  a  note  be  drawn,  payable 
in  3  mo.,  that  the  net  proceeds  may  be  $15000  after  deduct- 
ing the  bank  discount  at  8%? 


rEKCKNTAGE.  li  10 


REVIEW  EXERCISES. 

375.  1.  What  sum  must  l)e  invested  at  8^  to  yield  an 
annual  income  of  $1-10()? 

2.  A  merchant  bought  a  1)111  of  goods  amounting  to  $7825 
on  3  mo.  credit,  but  was  offered  a  discount  for  cash  of  4%. 
What  was  the  difference  in  the  offers,  money  being  worth 

3.  A  man  bought  a  farm  of  135  A.  25  sq.  rd.  for  $62.50 
per  acre.  He  pn'id  one-third  of  the  purchase  money  in  cash ; 
one-half  the  remainder  in  6  mo.,  and  the  balance  in  1  yr. 
3  mo.  Money  being  worth  6^,  what  would  have  been  the 
cjish  price  of  the  farm? 

4.  A  will  contained  a  bequest  of  $4500  to  a  charity  hos- 
pifal,  to  be  paid  in  1  yr.  3  mo.  Money  being  worth  7%, 
what  was  the  cash  value  of  the  bequest? 

5.  What  is  the  amount  of  $3752  for  3  yr.  2  mo.  15  da., 
with  compound  interest  at  7%? 

6.  $175.  Cincinnati,  O.,  March  25,  1872.     ' 
For  value  received,  on  demand  I  promise  to  pay  J.  H. 

Sheppard,    or  bearer,   One  Hundred   Seventy-five  Dollars, 
with  interest  at  8%.  j)^^^.,j,  j3  Aspell. 

This  note  was  paid  April  15,  1876.     How  much  was  due  ' 
upon  it? 

7.  What  is  the  difference  between  the  true  discount  and 
the  bank  discount  of  $5728  for  3  mo.  at  8%  ? 

8.  On  a  note  of  $3729.75,  dated  Feb.  20,  1872,  bearing 
6j^  interest,  were  the  following  indorsements;  July  15,  1872, 
$525;  Dec.  15,  1872,  $478;  Feb.  20,  1873,  $25;  May  17, 
1873,  $75;  Sept.  2«,  1.H73,  $1000.  Whnt  um<  .1.,..  .T,...,  15. 
1874? 


246  PERCENTAGE. 

9.  A  coal  dealer  bought  8790  T.  of  coal  at  $3.75  per  T. 
He  sold  15 fo  of  it  at  a  gain  of  10^  on  the  cost,  40%  .1  it 
at  a  gain  of  5%,  and  the  rest  at  a  gjiin  of  8^.  He  i)ai(l 
3}-^  of  the  cost  for  traiif>iv>rratio!).  How  luiich  did  he  ^^ain 
by  tlic  sjKiculatiou  ? 

10.  A  grain  specula....  ^.m.^ui  ^w,,,,,*  uuMirl.s  ol'  Imrlcy,  at 
85  cents  per  bii«hcl  cash.  He  sold  it  the  same  day  at  an  ad- 
vance of  4% ,  receiving  in  payment  a  note  due  in  1  mo.,  wliich 
he  had  discounted  at  a  bank  ••♦  *''  What  was  his  jrain  in 
cash? 

11.  A  man  bought  a  house,  p«iynig  62^^  of  the  price  in 
cash,  and  the  rest  in  notes  to  tho  nniount  of  9-":iu)  What 
was  the  cost  of  the  house? 

12.  A  merchant  sold  15^  of  his  st<x-k  of  dry  goixls  the 
first  month,  10^  the  second  month  and  25^  of  the  remainder 
the  third  month,  when  he  took  an  inventory  of  his  stock  on 
hand,  and  found  that  he  had  remaining  85300  worth  of  goods. 
AVhat  was  the  original  value  "of  the  stock? 

13.  What  number  is  that  to  which,  if  f  of  25%  of  J  of  480 
be  added,  the  sum  will  equal  25%  off  of  50%  of  324? 

14.  A  speculator  bought  1000  bbl.  of  flour  at  a  given 
price  per  bbl.  paying  J  of  its  value  in  cash  and  giving  a  bank- 
note for  2  mo.  for  the  balance,  which  was  discounted  on  the 
day  it  was  given  at  6%.  The  discount  on  the  note  was  $31.50. 
How  much  did  he  pay  for  the  flour  per  bbl.  ? 

15.  What  is  the  difference  between  the  simple  and  the  com- 
pound interest  of  84725.50  for  2  yr.  2  mo.  15  da.  at  6%. 

16.  What  is  the  difference  between  the  amount  of  83240 
for  5  yr.  3  mo.  10  da.  at  7%  simple  interest,  and  the  amount 
of  the  same  sum  for  the  same  time  and  rate,  with  interest 
payable  annually? 

17.  A  gentleman  invested  \  of  his  annual  income  in  mort- 
gages, paying  6^  annual  interest.  In  6  mo.  12  da.  his  in- 
terest from  them  was  8640.    What  was  his  annual  income? 


rUOIll     AND    L(J6S.  247 


PROFIT  AND  LOSS. 

376.    1.  AVheu  25^  is  gained  what  part  is  gained? 

2.  I  sold  a  coat  which  cost  nie  ?12  for  25^  more  than  it 
cost.     How  much  did  I  gain  ?     How  much  did  I  get  for  it? 

3.  Paid  315  for  a  ton  of  hay,  and  sold  it  at  a  loss  of  20^. 
How  much  did  I  lose?     How  much  did  I  get  for  the  hay? 

4.  If  I  sell  land  that  cost  me  850  an  acre,  at  an  advance  of 
10^ ,  how  much  do  I  get  per  acre  ? 

5.  If  I  paid  S50  per  acre  for  ray  land  and  sell  it  at  $5  per 
acre  less  than  I  paid,  what  part  of  the  cost  do  I  lose  ?    What  %  ? 

6.  If  I  sell  flour  for  38  per  bbl.  that  cost  me  36,  what  i)art 
of  the  cost  do  I  gain  ?    AVhat  per  cent.  ? 

'  7.  Sold  boots  at  35  a  pair  that  cost  34,  what  part  of  the  cost 
was  gained  ?     What  per  cent.  ? 

8.  By  selling  flour  at  a  profit  of  32  per  bbl.,  20^,  or  |,  of 
the  cost  was  gained.     What  was  the  cost? 

9.  Sold  wheat  at  a  profit  of  3 .10  2>er  bu.  which  was  5^  of 
its  cost.     What  was  its  cost? 

10.  If  I  sell  a  cow  that  cost  me  350,  at  an  advance  of  20^ 
on  the  cost,  how  much  will  my  profit  be  ?  How  much  do  I 
get  for  her? 

11.  By  selling  flour  at  310  per  bbl.,  a  profit  of  25^  was 
lua.l       What  did  it  cost? 

Analysis. — Since  25^  or  \  of  the  cost  was  gained,  the  selling  price 
must  Imve  been  J  of  the  cost;  and  since  J  of  the  cost  was  $10,  J  of  the 
cost  is  \  of  $10,  which  is  $2 ;  and  since  |^  of  the  cost  is  $2,  the  cost  is  4 
times  $2  or  $S. 

12.  By  selling  dress  goods  at  66  cents  })or  yl  ,  a  profit  of 
10^  was  made.     What  was  the  cost '! 

13.  By  selling  tea  at  $.80  a  poun«l,  u  l<.s>  .  t  l  i  uas  in- 
curred.    What  was  the  cost? 


248  PERCENTAGE. 

14.  K  I  get  $60  for  a  cow,  and  thereby  gain  20^ ,  or  J  of 
the  cost,  what  did  she  cost  me? 

15.  I  bought  a  horse  for  $150  and  sold  him  at  uii  advance 
of  20^ .     What  did  I  get  for  him  ? 


DEFINITIONS. 

377,  I^ro/it  and  Loss  are  terms  used  to  denote  the  gain 
or  loss  in  businesjj. 

378.  The  processes  in  Profit  and  Loss  involve  the  same 
elements  as  do  the  fundamental  problems  in  Percentage.  The 
corresponding  terms  are: 

1.  The  Cost,  or  Sum  Investrd,  is  the  Jiase, 

2.  The  Rate  Per  Cent,  of  profit  or  loss  is  the  Ttatc. 

3.  The  Gain  or  I/m  is  the  ^Percentage, 

4.  The  Selling  PricCy  when  more  than  the  cost,  is  the 
Amount. 

5.  The  SeUiwj  I^rir, ,  when  less  than  the  cost,  is  the 
Difference, 

Principle. — The  gain  or  loss  is  reckoned  at  a  certain  rate  j^er 
cent,  on  the  cost  or  sum  invested. 


WBITTES    EXERCISES. 

379.    1.  A  paid  $3500  for  a  house,  and  sold  it  at  10^  ad- 
vance.   How  much  did  he  gain?    How  much  did  he  get  for  it? 

PROCESS.  Analysis. — Since  the 

$3500  X     .  10  ==  $  350,  Gain.  '^«"«^  ^'^  ''^^^  ^^  «"  ^^- 

vance  of  lOffr  on  the  coat, 

$3500  +  $350  =  $3850,  Selling  price,     his  gain  was  ■,\%  or  J^ 
Cost  X  Rate  =  Gain.  of  $8500,  Avhich  is  $350; 

and  the  selling  price   is 
equal  to  the  sum  of  the  cost  and  gain,  or  $3850. 


PROFIT   AND   LOSS.  249 

Rules. -/Si/ice  Uw  savie  elemetits  are  involved  as  in  the  Jun- 
damental  jyroblenid  in  Percentage  Uie  rules  are  ilie  sanie. 

FORMULAS. 

1.  Gain  or  loss  =  Cost  X  Rate. 

2.  Rate  =  Grain  or  loss  -^  Cost. 

3.  Cost  =  Gain  or  loss  ~-  Rate. 

4.  Cost  =  Selling  price  ~  (  1  -f  Rate  ). 

5.  Cost  =^  Selling  price  -r-  (  1  —  Rate  ). 

2.  Mr.  A.  bought  a  piano  for  $275,  and  sold  it  at  an  ad- 
vance of  25 J^.     How  much  did  he  receive  for  it? 

3.  A  bookseller  bought  83584  worth  of  books,  and  sold 
them  at  a  gain  of  10^.  .  How  much  did  he  gain? 

4.  A  carriage  maker  sold  a  carriage  at  25^  advance  on  the 
cost.     It  cost  him  3318.25.     How  much  did  he  get  for  it? 

5.  A  harness  maker  sold  a  set  of  double  harness  at  a  profit 
of  Ibfc.     It  cost  him  $45.     What  did  he  get  for  it? 

6.  A  manufacturer  of  tools  sold  5  dozen  axes  at  a  profit  of 
12^.     They  cost  him  $9  jxjr  dozen.     What  was  his  profit? 

7.  A  merchant  sold  a  bill  of  goods  at  a  profit  of  15^. 
The  goods  cost  him  $84.25.     What  did  he  receive  for  them? 

8.  A  speculator  bought  50000  pounds  of  sole  leather,  which 
he  sold  at  a  profit  of  8^.  If  it  cost  him  $6000,  what  did  he 
get  for  it? 

9.  A  man  sold  his  house  at  a  profit  of  15^.  If  he  paid 
$3000  for  it,  how  much  did  he  get  for  it? 

10.  A  drover  sold  a  flock  of  sheep  at  a  piufil  ul"  7^/. 
If  they  cost  him  $1500,  what  did  he  get  for  them? 

11.  A  |H)ultry  dealer  liought  a  quantity  of  poultry,  wliich 
ho  sold  at  a  gain  of  O'/T.  He  paid  8250  for  it.  How  much 
did  he  get  for  it? 

12.  Mr.  A  lK)Uglit  cloth  at  $2.15  ])er  yard.  At  what  price 
must  lie  sell  it  t(»  gain  30^,  ? 


250  PERCENTAGE. 

i  !.  A  farm  which  cost  $65  per  acre  was  sold  at  a  gain  of 
15 J^.     For  how  much  did  it  sell  per  acre? 

14.  A  merchrfut  bought  31)50  yards  of  cotton  at  9J  c^nts 
a  yard.     How  much  will  he  get  for  it  if  he  sells  it  at  a  gain 

of  l2i;^? 

15.  A  merchant  desires  to  mark  goods  that  cost  him  $3.60 
per  yard  so  that  he  may  gain  33J^.  At  what  price  must 
the  goods  be  marked? 

16.  A  bankrupt  .«took  was  sold  at  35^  loss.  What  was 
the  selling  prici  i  articles  that  cost  50c.?  $1?  $1.50? 
$1.75? 

17.  What  jxjr  cent,  is  lost  by  selling  sugar  at  10  cents  per 
pound  which  cost  12  cents  per  pound? 

PROCESS.  Analysis.-t- Since   sugar   that  cost   12 

A   J2 ft  10  =  fi  0^        cents  was  sold  for  10  cents,  there  was  a 

loss  of  2  cents  jjcr  |)Ound.  And  since  the 
V  'O^  -r-9,iZ  =  lo^y^  gjijn  or  loss  is  reckoned  at  a  rate  per  cent. 
Loss  -4-  Cost  =  I^itc.      on  the  cost,  we  must  tind  what  per  cent.  2 

is  of  1  '   t'f  I'J;    or,  expressed  as 

hundredths,  is  .IGf  of  12,  or  1G^<, . 

18.  What  |)t  r  win.  is  gained  by  selling  tea  at  81  which 
cost  $.75? 

19.  What  per  cent,  is  lost  by  selling  tea  at  $.75  which 
cost  $1? 

20.  What  per  cent,  is  lost  by  selling  cloth  at  $1.25  that 
cost  $1.75? 

21.  Bought  goods  at  50  cents  a  yard  and  sold  them  at  60 
cents  a  yard.     What  per  cent,  did  I  gain? 

22.  Goods  that  are  selling  at  12^  cents  a  yard  cost  10  cents. 
What  per  cent,  is  gained  by  selling  them  at  that  rate? 

23.  A  man  bought  a  city  lot  for  $4500  and  sold  it  for 
$.5000.     AVhat  per  cent,  did  he  gain  ? 

24.  Sold  a  quantity  of  potatoes  for  $850  which  cost  me 
$970.     What  im-  cent,  did  I  lose? 


PROFIT   AND    LOSS.  251 

25.  Bought  a  quantity  of  crude  petroleum  at  5  cents  per 
gallon  and  sold  it  at  4J  cents.    What  per  cent,  did  I  lose? 

26.  A  fruiterer  bought  10  lx)xe3  oranges  at  61.75  per  box. 
Two  of  the  boxes  were  worthless,  but  he  sold  the  balance  at 
such  price  that  he  gained  5^  on  the  whole  purchase.  How 
much  did  he  sell  them  for  per  box?  How  much  did  he  gain 
on  the  purchase? 

27.  I  bought  books  at  10^  discount  from  the  retail  price, 
which  was  81.50  jMjr  volume,  and  sold  them  at  the  retail 
price.     What  was  my  gain  per  cent.  ? 

28.  An  agent  gets  a  discount  of  40 J^  from  the  retail  price 
of  articles  and  sells  them  at  the  retail  price.  AVluit  is  his 
gain  per  cent.  ? 

29.  A  merchant  bought  cloth  at  ^8.25  per  yard,  and  after 
keeping  it  6  months  sold  it  at  83.75  jxjr  yard.  What  was 
his  gain  per  cent.,  reckoning  6^  |xjr  annum  for  the  use  of 
money  ? 

30.  Which  is  more  profitable,  and  how  much  per  cent.,  to 
sell  goods  for  cash,  at  once,  at  25j^  advance,  or  in  1  year  at 
30^  advance,  money  Ixjing  worth  7^  ? 

31.  Mr.  A.  gets  a  discount  of  30^  from  the/retail  or  list 
price  of  goods.  Mr.  B.  gets  a  discount  of  30^  also,  and  5% 
off  for  cash.  If  both  sell  goods  at  the  list  price,  Avhat  is  each 
one's  gain  i)er  cent.  ? 

32.  A  merchant  bought  tea  at  20j^^  k«6  than  its  markc  t 
value,  and  received  a  discount  of  5^  for  cash.  He  sold  it 
at  an  advance  of  15^  above  its  market  value.  What  was 
his  gain  per  cent.  ? 

33.  By  selling  cloth  at  a  gain  of  12  cents  a  yard,  I  real- 
ized a  gain  of  8^  on  the  e^st.     What  was  the  cost? 

PK0Ct:ss.  .\naly8IS, — Since  the  gain,  12  cuit.-.  is 

0  12—   08    -?t1.50.      ^/^'  **'■  T*v  o^  the  cost,  I  of  12  wnts,  or 
1  \  cents,  ia  iJj  of  the  cost,  an<l  tht>  cost  is 

(iaiii        Ual  •  100  times  ij  cent^  or  $1.50. 


252  PERCENTAGE. 

1  L    I   make  10^   by  selling  tea  at  a  profit  of  10  cents  a 
jx)Uiul.     What  did  it  cost  me?     What  do  I  sc^ll  it  fur? 
.    35.  Flour  was  sold  at  a  profit  of  $1.50  jki-  1)1)1.,  wlildi  was 
16Jjg  of  the  cost.     What  was  the  cost  / 

36.  A  merchant  made  12j^  by  selling  cloth  at  an  advance 
of  12  cents  a  yard.     What  did  it  cost? 

37.  By  selling  butter  at  8  cents  a  pound  more  than  cost,  a 
grocer  made  20^.     What  did  he  j)ay  for  it? 

38.  A  merchant  sold  cloth  which  was  damaged  by  fire,  at 
a  sacrifice  of  22  c^nts  per  yard,  which  was  40^  of  the  cost. 
AVliat  did  the  goods  cost? 

39.  A  farmer  sold  a  yoke  of  cattle,  to  which  he  had  fed 
$10  worth  of  grain,  at  an  advance  of  $25,  and  still  realized 
a  profit  of  15 fc     What  did  they  cost? 

40.  A  man  sold  a  horse  at  an  advance  of  $75,  which  was 
a  gain  of  25^.     What  was  the  cost  of  the  horse? 

41.  If  I  sell  a  quantity  of  apples  at  an  advance  of  25  cents 
a  barrel,  and  thereby  realize  12 J^  profit,  what  was  the  cost? 

42.  By  selling  cloth  at  a  gain  of  23  cents  per  yard,  I  realize 
a  profit  of  20^C'     What  did  it  cost? 

43.  A  merchant  asked  25%  more  for  his  goods  than  they 
cost  him,  but  at  last  sold  them  at  a  reduction  of  10^  from 
his  asking  price,  thus  realizing  from  the  sale  $4684  profit. 
What  was  the  cost  of  the  goods  ? 

44.  A  gentleman  sold  a  horse  for  $180  and  gained  20^  on 
him.     What  did  the  horse  cost? 

PROCESS.  Analysis. — Since  20^  of 

1  0  0  %  +  2  0  ^  =  1  2  0  ^  the  cost  w»8  gained  the  «ell- 

^'^    '  '^  ^^  ing  price  must  liave  been  20^ 

$180-7-1. 20  =  $150  more  than  the  cost,  or  120fo 

Selling  Price  -r-  (1 4-Kate)  =  Cost,    of  the  cost.    And,  since  1 20 fo 

of  the  cost  is  $180,  Ifc  of  the 
cost  is  y^^  of  $180,  or,  $1.50,  and  the  whole  cost  is  100  times  $1.50, 
or  $150. 

Therefore  the  horse  cost  the  gentleman  $150. 


ri:ui  11    AND  LOSS.  253 

45.  A  gentleman  soltl  a  carriage  for  S230,  and  thereby 
lost  8%  of  the  cost.     What  was  the  cost? 

I'KOCESS.  Analysis, — Since  8^  of  the 

-i  Of)  c/  k  (/ 92^  ^^^^  ^^^  ^^^'  ^^^  selling  price 

/o  yo  /o  ^^^^  1^^^.^  ^^^  g^  j^^  jIjj^j^ 

$  2  3  0  -I-  .  9  2  =  $  2  5  0  the  cost,  or  92%  of  the  cost. 

Selling  Price -^(1— Rate)  =  Cost.     And,  since  92/^  of  the  cost 

was  $230,  l^c  of  the  cost  was 
^  of  $230,  or  $2.50,  and  the  whole  cost  was  100  times  $2.50,  or  $250. 

46.  By  selling  apples  at  6  .50  per  bushel  a  grocer  gained 
25%  on  the  cost.     AVhat  was  the  cost? 

47.  A  farm  was  sold  for  S38000,  which  was  a  loss  of  5^ 
of  the  cost.     What  was  the  cost? 

48.  A  block  of  stores  was  sold  for  ^185000,  which  was  a 
gain  of  15 fc     What  did  they  cost? 

49.  A  merchant  lost  5^  by  selling  calico  at  9 J  cents  a 
yard.     What  did  it  cost  ? 

50.  A  bankrupt  stock  was  sold  for  $3582,  which  was  a  loss 
of  33 i^.     What  did  it  cost? 

51.  By  selling  molasses  at  65  cents  a  gallon,  a  LTocor 
gained  30 J^.     AVhat  was  its  cost? 

52.  A  man  was  comiK'lled  to  sell  his  household  furniture 
for  81250,  which  was  a  loss  of  37 1^.     What  did  it  cost? 

53.  A  boot  and  shoe  dealer  lost  9^  by  selling  boots  at 
$3.75  a  pair.     AVhat  was  the  cost  of  the  boots? 

54.  A  stationer  lost  26^  by  selling  i)aper  at  $2.22  a 
ream.     What  did  he  pay  for  it? 

55.  A  druggist  gained  125^  by  selling  alcohol  lor  $3.50 
per  gallon.     What  did  he  pay  for  it? 

56.  Coal  was  sold  at  $4.56J^  per  ton,  which  was  a  loss  of 
17^.     What  was  the  cost? 

57.  When  pork  is  selling  at  $4.50  per  hundred-weight  I 
loee  105^.  What  will  be  my  gain  ixr  <•  in.  if  T  sell  at  $6 
per  hundred-weight? 


264  PERCENTAGE. 


COMMISSION. 

liSO.  1.  If  a  man  gells  $500  worth  of  goods  for  me,  how 
much  will  he  receive  if  he  gete  2^  of  the  sales? 

2.  If  I  allow  a  man  2^  for  purchasing  $3000  worth  of 
silks  for  me,  how  much  will  he  get  for  his  services? 

3.  If  I  collect  a  debt  of  $350,  and  charge  4^  of  the 
amount  for  my  services,  how  much  will  I  receive? 

4.  Mr.  B.  paid  his  agent  5^  for  selling  83000  worth  of 
cotton.  How  much  did  he  pay  him,  or  what  was  his  com- 
mission t 

5.  At  3^  commission,  how  much  will  A  receive  for  sell- 
ing $4200  worth  of  flour?  How  much  will  lie  left  after  pay- 
ing the  commission,  or  what  will  be  the  net  proceeds  f 

6.  If  I  pay  2^  commission  for  buying  goods,  what  is  the 
cost  of  every  dollar*s  worth  of  goods  bought?  Since  every 
dollar's  worth  of  goods  bought  costs  the  purchaser  $1.02,  how 
many  dollars'  worth  can  be  bought  for  8102?     For  $204? 

7.  If  I  pay  3^  commission  for  buying  goods,  how  many 
dollars'  worth  of  goods  ciin  be  bought  for  $309,  after  paying 
the  commission?     For  $515? 

8.  How  many  dollars'  worth  of  goods  can  be  purchased 
for  $630,  after  making  allowance  for  the  agent's  commission 
at  5^  of  the  value  of  the  goods  purchased? 

DEFINITIONS. 

381.  A   Cotnniission  Merchant  or  Agent  is  a 

person  who  buys  or  sells  goods,  or  transacts  other  business  for 
another. 

382.  The  Conitnission  is  the  compensation  or  percent- 
age allowed  a  commission  merchant  or  agent. 


COMMISSION.  255 

A  Consif/nnient  is  a  quantity  of  merchandise 
sent  to  a  cominisiiion  inerchaut  or  agent  to  be  sold. 

384.  The  Consignor  is  the  person  who  sends  the  mer- 
chandise to  be  sold. 

385.  The  Consignee  is  the  person  to  whom  the  mer- 
chandise is  sent. 

380.  The  Net  I^roeeeds  of  a  sale  is  the  snm  left  after 
the  commission,  expenses,  etc.,  have  been  deducted. 

387.  The  processes  in  Commission  involve  the  same  ele- 
ments as  do  the  fundamental  problems  in  Percentage.  The 
corresponding  terms  are: 

1.  The  Sales  or  Sum  Invested  is  the  IBase, 

2.  The  Rate  Per  Cent,  is  the  Hate. 

3.  The  Commission  is  the  Percentage, 

4.  The  Purchase  Price  plus  the  Commission  is  the  Amount, 

5.  The  Nd  Proceeds  is  the  Difference. 

388.  Principle. — The  commission  is  reckoiied  at  a  certain 
rate  per  cent,  on  the  value  of  Vie  sales  and  purduises. 

w  T!  r  T  T  i:  V     /:  \-  r  tj  r  t  s  t:  s  . 

389.  1.  What  will  be  an  agent's  commission  for  selling 
$385.15  worth  of  goods  at  3}J^  ? 

PROCESS.  Analysis. — Since  the  rate 

$385.15X.03i  =  $13.48  of  co,nmis«ioni8  3t/„or.a3}, 

*  the  ooinmissiun  will  be  .03^ 

fides  X  Kate  =  Commission.  „f  $385.15,  which  l»  $13.48. 

2.  What  is  the  commission  for  selling  cattle  to  the  value 
of  83184  at  2}^? 

3.  What  is  the  commission  for  selliog  cotton  to  the  value 
of  $8000,  at  2 J  i>er  rent.  ? 


25C  PERCENTAGE. 

4.  Mr.  B.  senfhis  agent  $3468  to  invest  in  goods,  allowing 
him  2^  commission.  What  sum  did  he  invest  in  goods  after 
deducting  his  commission  ?    What  was  the  agent's  commission? 

PROCESS.  Analysis.— 

$3468-^1.02  =  13400,  Amount  invested.     ^^"^*^  *^^  .T"' 

gets  a  commission 
Jlemittance  -i-  (1  +  Rate)  =  Purchase  Price.     „{  2^  for  purchas- 

$3468  — $3400  =^$68,  a)mmis8ion.  ing,  it  requircB 

$1.02  to  purchase 
$1  worth  of  good»;  he  can  therefore  ptirchaBc  as  many  dollars'  worth 
of  goods  for  $3468  aa  $1.02  is  contained  times  in  $3468,  which  h  34(X) 
timeA.  Therefore,  he  can  purchase  $3400  worth  of  goods.  The  money 
sent  minus  the  amount  invested  will  be  the  commission. 

Rules. — Since  the  tame  elemenU  are  involved  as  in  tiie  fun- 
damental problems  in  Percentage^  Hie  rules  are  the  same. 

FORMULAS. 

1.  Commission  =  Sales  oi*  Purchase  X  Rate. 

2.  Rate  =  Commission  -i-  Sales  or  Purchase. 

3.  Sales  or  Purchase  =  Commission  -h  Rate. 

4.  Purchase  =  Sum  remitted  -r-  (  1  -f  Rate  ). 

5.  Sales  =  Net  Proceeds  -^  ( 1  —  Rate  ). 

5.  If  I  send  my  agent  $4050  to  invest  in  goods,  after  de- 
ducting 3^  commission,  what  sum  will  he  invest? 

6.  If  I  send  my  agent  $875  to  invest  in  calico,  allowing 
him  2^  commission,  how  many  yards  can  he  buy  at  6  cents 
per  yard? 

7.  How  much  is  an  agent's  commission  for  selling  385  bbl. 
flour  at  $6.50  per  bbl.,  the  rate  of  commission  being  2^  ? 

8.  What  is  the  commission  for  collecting  bills  to  the 
amount  of  $784.25  at  5^  ? 

9.  How  much  must  an  agent  be  paid  for  selling  25  bbl. 
pears  at  $12.75  {)er  bbl.,  his  commission  being  6^  ? 


COMMISSION.  257 

10.  Wlisit  k  the  commission  at  3^  for  selling  125  bbl. 
of  |K)tat()cs  at  $2.37  J  per  bbl. 

11.  A  commission  merchant  sold  20  firkins  of  butter,  each 
containing  56  lbs.,  at  28  cents  per  lb.  How  much  was  his 
e()mmi.ssion  at  8^  ? 

12.  What  is  the  commission  at  7^  on  a  sale  of  20  boxes 
of  eggs,  each  containing  22  doz.,  at  23  cents  per  doz.? 

13.  How  much  commission  must  be  paid  to  a  collector  for 
collecting  an  account  of  S928.75  at  3|^? 

14.  I  sent  my  agent  $1525  to  be  invested  in  goods  after 
deducting  his  commission  of  2^.     How  much  did  he  invest? 

15.  A  merchant  sent  his  agent  8375.50  to  invest  in  muslin 
at  8  cents  jx^r  yard.  After  deducting  3^  commission,  how 
many  yards  of  muslin  did  he  purchase? 

16.  Mr.  A.  sent  33320.10  to  be  invested  in  goods  after  pay- 
ing his  agent  2^  commission.     What  sum  was  invested? 

17.  A  man  sent  his  agent  $3725.05  to  invest  in  pork  after 
deducting  H^  commission.     How  much  did  he  invest? 

.18.  A  speculator  sent  his  agent  in  Chicago  $8966.75  to  in- 
vest in  wheat.  After  deducting  }^  commission,  how  many 
l)ushe]s  of  wlieat  did  he  buy  at  $1.11|  per  bushel?  What 
was  the  commission? 

19.  An  agent  received  $24.52  for  selling  good*  at  a  toni- 
iirission  of  2^.  How  many  dollars' worth  of  goods  did  he 
sell? 

20.  How  much  money  must  I  send  my  agent,  so  that  he 
may  purchase  for  me  150  bbl.  flour  at  $8.25  per  bbl.  if  I  jwiy 
him  3^  commission  for  his  services? 

21.  A  commi.ssion  merchant  received  $318.25  for  selling 
$12730  worth  of  bankrupt  goods.  What  was  the  rati  .f 
cornmi.ssion? 

22.  A  sale  of  real  estate  returned  as  net  proceeds  $2396.49, 
utter  paying  $324.18  charges  and  a  commission  of  2^.  For 
liow  much  did  it  sell? 

17 


258  PERCENTAGE. 


REVIEW  EXERCISES. 

300.  1.  A  man  whose  wages  had  been  $27  per  week  was 
obliged  to  take  33 J^  less.     How  much  was  the  reduction? 

2.  A  man  bought  a  horse  for  $300  and  sold  hini  for  $375. 
What  per  cent,  did  he  gain? 

3.  When  sugars  that  cost  10  cents  a  pound  are  sold  for  11 
cents,  what  per  cent,  is  gained? 

4.  When  land  is  selling  at  an  advance  of  840  an  acre,  what 
is  the  gain  per  cent,  if  it  cost  8120  an  acre? 

5.  A  l)oy  sold  apples  at  the  rate  of  2  for  3  ctnU  which  he 
bought  at  the  rate  of  3  for  2  cents.  What  did  he  gain  per 
cent.  ? 

6.  A  nev...  ci^ciii  sold  831  wurth  of  goods  in  a  day  at  a  com- 
mission of  10^.     How  much  w.as  his  commission? 

7.  A  man  lost  680  which  was  just  20^  of  Jiis  money.  How 
much  money  had  he? 

8.  A  man  paid  2^  for  selling  his  wheat  and  realized  S1..47 
per  bushel.     For  how  much  did  it  sell  per  bushel? 

9.  The  interest  on  8240  for  2  vPnr>  was  828.80.  At  what 
per  cent. was  it  loaned? 

10.  When  money  is  loaned  at  6^^  and  the  interest  amounts 
to  875  on  81000,  how  long  has  it  been  loaned? 

\  11.  When  a  merchant  buys  goods  at  ^  of  their  estimated 
value,  and  sells  them  at  their  estimated  value,  how  much  is 
his  gain  per  cent.  ? 

^    12.  When  a  man  sells  goods  at  a  price  from  which  he  re- 
N^ived  a  discount  of  30^,  what  is  his  gain  per  cent.? 
"^  13.  When  a  man  can  borrow  money  at  8^ ,  which  is  more 
profitable,  and  how  much  per  cent.,  to  buy  goods  at  3^  off 
for  cash  or  on  3  mo.  credit? 

^  14.  If  I  sell  a  horse  for  8125  which  cost  me  8175,  what 
do  I  lose  i^er  cent.  ? 


REVIEW   EXERCISES.  259 

15.  A  man  sold  a  cow  at  an  advance  of  810,  which  was 
25^  of  what  she  cost.     How  mucli  did  she  cost? 
>  16.  A  boy  can  pick  33  J^  as  many  apples  as  his  father. 
If  his  father  can  pick  18  barrels  a  day,  how  many  can  the 
hoy  pick? 

17.  When  a  merchant  buys  goods  at  a  discount  of  20^ 
from  the  regular  price,  and  sells  them  at  20^  more  than  the 
rei^ular  price,  what  is  his  gain  per  cent.  ? 

18.  Which  is  more  i)rofi table,  to  sell  goods  iioWj  that  cost 
18  cents  a  yard,  for  20  cents  a  yard,  or  to  keep  them  1  year 
and  sell  them  at  21  cents  when  money  is  worth  6^  ?  How 
much  more  profitable  on  an  investment  of  $1000? 

19.  Mr.  A.  l)ought  a  horse  and  carriage,  paying  twice  as 
much  for  the  horse  as  the  carriage.  He  afterward  sold  the 
horse  for  25^  more  than  he  gave  for  it,  and  the  carriage 
for, 20^  less  than  he  gave  for  it,  receiving  for  both  8577.50. 
What  was  the  cost  of  each  ? 

20.  After  getting  a  note,  without  interest,  discounted  at  a 
bank  for  3  months  at  7^,  I  had  $468.39.  What  was  the 
fiice  of  the  note? 

21.  A  man  can  borrow  money  at  6^  and  pay  cash  for 
goods,  obtaining  a  discount  of  2^,  or  he  may  pay  for  the 
goods  in  2  mo.  Which  is  the  more  advantageous,  and  how 
much,  on  an  invoice  of  gocxls  amounting  to  $1500? 

22.  I  had  a  note  for  $1000  discounted  at  a  bank  for  3 
months  at  7^.  The  proceeds  were  invested  in  wheat  at 
$l.r>5  per  bu.     How  many  bushels  did  I  buy? 

23.  Mr.  A.  sold  a  horse  for  8198,  which  was  10^  less  than 
he  aske<l  for  hiin,  and  his  asking  price  was  10^  more  than 
the  horse  cost  him.     What  was  the  cost  of  the  horse  ? 

24.  I  bought  a  horse  of  Mr.  A.  for  20^  less  than  he  cost 
him,  and  I  immediately  sold  the  horse  for  25j?^  more  than  1 
paid  for  him,  gaining  825.    What  did  thfe  horse  cost  Mr.  A.? 


260  PERCENTAGE. 

27).  1  iliroctcd  iijy  u^rcni  to  puivluihc  lor  mc  30  village  lots 
at  $050  each,  and  to  pay  the  expenses  of  examining  the  titles, 
which  averaged  34.25  j)er  lot.  What  did  they  all  cost  nie  if 
the  agent's  com  mission  was  4^  on  the  price  of  the  lots? 

26.  For  how  much  must  the  lots  be  sold  to  give  me  a 
net  profit  of  20^,  besides  allowing  tlie  agent  5^  cogimission 
for  selling? 

27.  After  a  certain  time  a  sum  of  money  which  had  been 
at  interest,  had  increased  18}^  and  amounted  to  $3896.74. 
What  was  the  sum  at  interest? 

28.  A  clothier  sold  two  suits  of  clothes  at  872  each.  ( )n 
one  he  gj\ined  20^,  and  on  the  other  he  lost  20^.  Did  he 
gain  or  lose  on  the  sale  and  how  imicli?  How  much  per 
cent.? 

2(V  After  marking  goods  at  an  advance  of  25^  over  cost, 
a  merchant  made  an  abatement  of  20j^  from  the  marked 
price.     Did  he  gain  or  lose,  and  how  much  per  cent.  ? 

30.  On  i  of  my  pro|)erty  I  gained  33J<J^  and  sold  the  rest 
for  J  of  the  cost  of  the  whole,  receiving  in  payment  a  note  due 
in  3  months  which  I  got  discounted  at  a  bank  at  6^.  What 
was  my  gain  per  cent,  if  my  pro|)erty  was  worth  810000? 

31.  I  sent  ray  agent  87000  to  be  invested  in  wheat  at  81.15 
psr  bushel,  allowing  him  a  commission  of  3^  on  the  purchase. 
I  paid  for  storage  2  cents  j)er  bushel  per  month,  and  1^  per 
month  for  the  use  of  the  money.  After  3  months  my  agent 
sold  the  wheat  at  $1.33  per  bushel,  charging  2^  commission 
for  making  the  sale,  and  took  in  payment  a  note  for  the 
amount  for  1  mo.,  which  I  had  discounted  at  a  bank  at  9%. 
Did  I  make  or  lose,  and  how  much? 

32.  A  man  wishing  to  sell  a  horse  and  cow,  asked  three 
times  as  much  for  the  horse  as  the  cow,  but,  finding  no  pur- 
chaser, he  reduced  the  price  of  the  horse  20^ ,  and  the  price 
of  the  cow  10%,  and  sold  them  both  for  8165.  What  did 
he  get  for  each? 


REVIEW   EXERCISES.  261 

33.  I  owe  B  60]}%  of  the  amount  I  owe  A,  and  I  owe  C 
40%  of  the  amount  I  owe  B.  How  much  do  I  owe  each  if 
I  owe  B  880  more  than  I  do  C? 

\j  34.  I  lx>ught  a  quantity  of  coffee  at  28J^  cents  per  pound. 
Allowing  the  coffee  to  fall  short  5^  in  weighing,  and  lOC^  of 
the  sales  to  be  lost  through  l)ad  debts,  for  how  much  must  I 
sell  it  {x^r  pound  that  I  may  make  a  clear  gain  of  20%  on 
the  cost? 

i  35.  A  quantity  of  prints  was  sold  at  a  commission  of  2^ 
and  the  proceeds  invested  in  cambrics  purchased  at  3% 
commission.  The  commission  for  buying  the  cambrics  was 
8126.30.     How  much  did  the  prints  sell  for? 

36.  A  tailor  sold  a  suit  of  clothes  for  S46,  thereby  gaining 
15^.  He  sold  another  for  860,  and  lost  the  same  amount 
of  money  which  he  gained  upon  the  first  suit.  What  per 
cent,  did  he  lose  upon  the  last  suit  sold? 
,  37.  A  merchant  having  a  quantity  of  pork  asked  33 J ^ 
more  than  it  cost  him,  but  was  obliged  to  sell  it  12^-  })er 
cent,  less  than  his  asking  price.  If  he  received  $7  per  cwt., 
what  was  its  cost? 

38.  What  must  l)e  asked  for  apples  which  cost  83  per  bbl., 
that  I  may  reduce  my  asking  price  20^,  and  still  gain  20^ 
on  the  cost? 

3i).  A  merchant  sold  a  consignment  of  blankets  at  3% 
commission,  receiving  in  payment  a  bank-note  for  1  mo., 
which  he  had  discounted  at  6^.  He  then  invested  the 
procee<ls  in  wool  at  30  cents  per  lb.,  charging  }^  commis- 
sion for  buying  it.  His  commis^sion  for  purchasing  the  w(k)1 
was  845.  What  was  the  value  of  the  coiisitjnmont?  How 
much  wool  did  he  buy? 

40.  A  merchant  sold  a  qiiuiim  >  i-i  u-m-m.-  ch  .i  ua.i.  »-»  20^. 
If,  however,  he  ha<l  purchajfetl  the  goods  for  860  less  than  he 
did,  hb  gain  would  have  been  25^.  What  did  the  goods 
co:-t? 


262  l»KKCENTAOE. 


TAXKS. 

CASE  I. 
GENERAL  TAXES. 

'.l\H,  1.  If  ft  pcrifon  has  to  jwy  nnnimlly,  for  public  pur- 
jH.M's,  2^  of  the  value  of  his  prr|KTty.  estimated  at  Sr)000, 
how  much  will  be  his  tnj-  f 

2.  If  I  am  taxed  l.J^  on  my  land,  houses,  etc.,  or  real 
estate,  estimated  at  $20000,  how  much  will  be  my  tax? 

3.  If  I  am  taxed  1%  on  the  value  of  my  movable  or 
penonal  property,  what  is  the  tax  on  $6000? 

4.  My  property  is  estimated  by  the  cissessors  to  be  worth 
$40000.*    Wlittt  will  be  my  tax  at  li%? 

5.  I  imported  500  yards  of  silk,  invoiced  at  82  per  yard. 
What  will  be  the  Government  tax,  or  duty,  upon  it  at  35^  ? 

6.  I  imported  500  yards  carpeting,  invoiced  at  82  per 
yard.  What  will  be  the  duty  at  50%  of  the  value,  or  50^ 
ad  valorem  f 

7.  Mr.  A.  imported  5000  yards  sheeting.  What  will  be 
the  duty  at  4J  cents  per  yd.,  or  the  4J  cents  specific  duty? 


DEFINITIONS. 

392.  Real  Estate  is  fixed  property;  as,  houses,  lands, 
tenements,  etc. 

393.  Personal  Property  is  movable  property;  as 
money,  stocks,  mortgages,  cattle,  etc. 

394.  A  Tax  is  a  sum  of  money  assessed  upon  the  ])er- 
sons,  property,  income,  or  business  of  individuals  for  public 
pur|X)ses. 


TAXES.  263 

395.  A  Property  Taa*  is  a  tax  on  projx^rty.  It  is 
reckoned  at  a  certain  rate  per  cent,  on  the  estimated  or 
assessed  value  of  the  property. 

390.  A  Personal  Tax  is  a  tax  assessed  upon  the  per- 
son.    It  is  called  a  poll  or  capitation  tax. 

397.  An  Assessor  is  an  officer  appointed  to  estimate  the 
taxable  value  of  property  and  prepare  the  assessment  roU. 

398.  An  Assessment  Moll  is  a  list  of  the  names  of 
taxable  inhabitants,  and  the  value  of  each  person's  property. 

Before  taxes  are  assessed  a  complete  inventory  of  all  the 
taxable  property  must  be  made.  If  the  assessment  includes 
a  poll  tax,  then  a  complete  list  of  taxable  polls  must  be 
made  out. 

* 

WRITTEN    EXERCISES. 

399.  1.  A  village  is  to  be  taxed  $3756  on  property  as- 
sessed at  6854315.  The  number  of  polls  at  $1.50  is  350. 
A's  proj)erty  is  assessed  at  85000,  and  he  pays  5  jk)11s;  B's 
pro|)erty,  at  $7000,  and  he  pays  5  polls;  C's  property,  at 
$3425,  and  he  pays  3  polls.  What  will  be  tho  tax  on  $1, 
and  what  will  be  the  tax  of  each? 

PROCESS. 

$1.50X350  =  $625,  Amount  of  poll  tax. 

$3756  —  $525  =  $3231,  Amount  to  be  levied  on  property. 

$3231  ^$854315  =  .00378,  or  Zj'^^  mi  11a  on  $1. 

$5000 X. 00378=  $18.90,  A's  property  tax. 

$1.60  X  5  =  $  7.60,  A'8  poll  tax. 

$26.40,  A'rt  entire  tax. 
$7000 X. 00378=  $26.46,  IV«  property  tax. 
$  1.50  X  5  =  $  7.i>0,  BV  poll  tax. 


264 


PERCENTAGE. 


400,  A  Uible  like  the  following  is  commonly  made  by  those 
who  compute  the  taxes. 


ASSESSOR'S 

TABLE.    (iZaie,  .00378.) 

Prop. 

Tax. 

Prop. 

•Tax. 

Prop. 

Tax: 

Prop. 

Tax. 

Prop. 

Tax. 

$1 

$.0037 

$7 

$.026 

$40 

$.151 

$100 

$  .378 

$700 

$  2.646 

2 

.0074 

8 

.03 

60 

.19 

200 

.756 

800 

3.024 

3 

.011 

9 

.034 

60 

,28 

300 

1.124 

900 

S.402 

4 

.015 

10 

.038 

70 

.26 

400 

1.512 

1000 

3.78 

5 

.019 

20 

.076 

80 

.302 

500 

1.89 

2000 

7.56 

6 

.023 

30 

.111 

90 

.34 

600 

2.268 

3000 

11.34 

2.  Find  C*s  tax  in  example  No.  1,  using  the  table. 

PROCESS. 

Tnx  by  table  on  $3000  is  $11.34 
Tax  by  table  on  $  400  is  $  1.512 
Tax  by  table  on  $    20  is  $    .076 
Tax  by  table  on  $      5  is  $    .019 


$1.50X3  = 


$12,947,  C's  property  tax. 
$  4.50  ,  C's  poll  tax. 
$17,447,  C's  entire  tax. 


Rule. — Divide  Vie  sum  to  be  raised j  after  deducting  the  poll 
tax,  by  the  whole  amount  of  taxable  propei-ty,  and  the  quotient 
wiU  be  the  rate. 

Multiply  each  man^s  property  by  Hie  rate,  and  to  the  product 
add  the  poll  tax,  if  any,  and  the  »um  tvill  be  the  whole  tax. 

3.  In  a  certain  town  the  assessed  value  of  property  was 
$1250500,  on  which  a  tax  was  levied  of  85008.125.  The 
number  of  polls  was  425  at  $  .75  each.  A's  property  is 
valued  at  818500;  B's,  at  822180;  C's,  at  15200.  A  pays 
5  polls;  B,  8  polls;  C,  9  polls.     What  is  the  tax  of  each? 


TAXES.  263 

4.  At  what  rate  must  $595000  worth  of  property  l)e  as- 
sessed to  raise  a  tax  of  82587,  if  there  are  300  polls  at  $1.50 
each?    What  will  be  the  tax  on  property  valued  at  $3400? 

5.  In  a  town  containing  275  j)olls,  assessed  at  $1  each,  the 
assessment  roll  shows  that  the  taxable  property  is  valued  at 
$895970.  The  tax  is  $2310.90.  What  is  the  rate  of  taxa- 
tion ? 

6.  A  tax  of  $35000  is  assessed  upon  a  certain  town.  The 
valuation  of  the  taxable  property  amounts  to  $4506000,  and 
there  are  650  taxable  polls  assessed  at  $1.25  each.  What 
will  be  the  tax  on  property  assessed  at  $11000? 

7.  What  sum  must  be  assessed  to  raise  S3750,  l)esides  pay- 
ing 2^  for  collection  ?  What  would  be  the  taxable  valuation 
of  property  to  raise  that  sum  if  the  rate  were  .003275? 


CASE  II. 

DUTIES  OR  CUSTOMS. 

4-01.  J>uties,  or  Customs^  are  taxes  levied  l)y  ^'•ov- 
cniment  upon  imported  goods. 

402.  A  Specific  T>uty  is  a  fixed  tax  on  certain  articles 
without  regard  to  their  value. 

403.  An  Ad  Vfllovcm  Dttttj  is  a  tax  of  a  certain 
jKJr  cent,  on  the  net  value  of  the  articles  in  the  country  from 
which  they  .have  been  brought. 

404.  Tare  is  the  allowance  made  for  the  weight  of  a 
box,  bag,  etc. 

405.  Leakage  and  Hreahage  are  allowances  made 
for  leakage  and  breakage  during  transportation. 

400.  The  Dutie8^  or  Chtgtoms,  are  collected  by  oflScers  of 
the  Guverniiient  in  the  Cttsfnnt-hnttsrn, 


266  Ii;i:(  ENTAGE. 


WBITTSy   JBXERCJ8E8, 

407.  1.  What  is  the  duty  on  20  hhd.  molasses,  containing 
') ;  _;il.  each,  at  9  cents  per  gal.,  allowing  5^  for  leakage? 

PROCESS.  Analysis.— 

03  gal.  X    20=1260  gal.  gross.         t^2:^^1i 
12G0  gal.  X. 05  =  63  gal.  leakage.  the  groJamount, 

1260  gal.—  63  gal.  =  1  197  gaL  net     or  63  gal.,  arc  de- 

$  .09X1197  =  $!  n  7. 7n  duty.  ducted  for  leakage; 

and  since  the  duty 
on  1  g»l,  18  9  cents  L,'al.  it  is  1197  times  9  cents,  or  $107.73. 

2.  What  is  the  duty,  at  5  cents  a  pound,  on  3750  pounds 
of  cofiee,  allowing  5^  for  tare? 

3.  What  is  the  duty  on  500  pounds  of  raisins,  in  boxes, 
valued  at  10  cents  a  jwund,  allowing  15^  for  tare,  when  the 
rate  of  duty  is  6^  ad  valorem? 

4.  What  will  be  the  duty  on  83000  worth  of  merchandise 
if  the  rate  of  duty  is  15%  ad  valorem? 

5.  What  is  the  duty,  at  20jg  ad  valorem,  on  7  tons  of  steel, 
of  2240  lb.  each,  invoiced  at  17  cents  per  lb.? 

G.  II.  K.  Thurber  &  Co.,  of  New  York,  imported  from 
Haviina  75  hhd.  molasses,  63  gal.  each,  valued  at  8o  rents 
per  gal.;  125  hhd.  sugar,  500  lbs.  each,  valued  at  0  cents 
per  lb. ;  800  boxes  cigars,  valued  at  $8  per  lx)X.  7^  was 
allowed  for  leakage  on  the  molasses;  duty  on  same,  25^; 
tare  on  siiL^ar.  45  lbs.  per  hhd.;  duty  on  same,  30^;  duty 
on  cigars,  GOy^.     What  was  the  amount  of  duties  paid? 

7.  A  wine  merchant  imported  45  casks  slierry  wine,  valued 
at  $65  per  cask;  56  casks  Madeira  wine,  ^ahled  at  860  per 
cask;  38  casks  German  wines,  valued  at  $37  per  cask.  If  an 
allowance  of  4^  be  made  for  leakage,  what  will  the  duty  be 
at  45^  ? 


irrnnuffliiiiiii 


liminilllllllTTTTlrflftllllllllllllHllllll 


STOCKS 


piiiiiuiihiiiimiM^mimuniiiinimu^ 

408.    1.  Into  how  many  shares  win  8100000  capital  stock 
of  u  company  be  divided,  if  the  shares  are  $100  each? 
Shares  will  be  regarded  as  $100  each  unless  otherwise  specified. 

2.  How  much  of  the  capital,  or  capital  stock  does  a  man 
own  who  has  30  shares?     25  shares? 

3.  How  much  st<x;k  is  represented  by  a  certificate  entitling 
thejiolder  to  40  shares? 

4.  What  is  the  selling  price  or  viarkct  value  of  10  shares  of 
railroad  stock,  when  stock  is  selling  at  its  original  value  or 
at  part 

5.  What  is  ilio  iiuirket  vahie  of  10  shares  of  stock,  if  it  is 
sold  at  105%  of  the  original  value  or  5%  above  par? 

6.  What  is  the  market  value  of  10  shares  of  stock,  if  it  is 
sold  at  95^  of  the  original  value  or  5%  below  part 

7.  What  will  10  shares  of  st<K5k  cost  at  5%  above  par,  if 
I  pay  a  dealer  in  stocks  or  stock-broker  \%  of  the  j>ar  value 
of  the  stock  for  buying  it? 

8.  What  will  5  shares  of  stock  cost  at  5^  below  par,  if  I 
pay  a  broker  \^  for  buying  or  for  brokerage  t 

9.  What  is  the  value  of  15  shares  of  bank  stock  at  90^ 
of  its  par  value? 

10.  What  is  the  cost  of  50  shares  Chicago  &  Rock  Island 
R.  R.  stock  at  90%  of  its  par  value? 

11.  What  is  the  cost  of  100  shares  Chicago  &  Alton  R.  R. 
stock  at  95%  of  its  par  value? 

(287) 


268  PERCENTAGE. 

12.  What  is  the  cost  of  20  shares  Pacific  Mail  stock  at 
2Q}^%  of  its  par  value? 

13.  A  company  whose  capital  stock  was  $50000,  gained 
$5000  above  expenses,  wliich  was  divided  among  the  stock- 
holders. What  per  cent  of  the  capital  stock  was  the  amount 
divided,  or  what  was  the  diimieiidf 

14.  What  amount  will  a  man  receive  who  owns  20  shares 
of  stock,  if  a  dividend  of  5^  is  declared? 

15.  What  amount  will  a  man  receive-  \sii  .  lui^  oU  .-iuires 
of  stock,  if  a  (J^  dividend  is  declared? 

16.  A  company  whose  capital  stock  was  $50000,  lacked 
$5000  of  meeting  its  obligations.  What  per  cent,  of  the  stock 
was  the  deficiency? 

17.  If  the  deficiency  is  made  up  by  the  stockholders,  how 
much  will  be  the  assessment  on  a  stockholder  who  owns  20 
shares  of  the  al)ove  stock? 

18.  How  much  will  be  the  assessment  on  15  shares  of  the 
above  stock? 

19.  What  will  be  the  annual  income  on  a  written  obliga- 
tion or  hoiid  for  $5000  which  yields  6^  interest  annually? 


DEFINITIONS. 

409.  A  Company  is  a  number  of  persons  associated 
together  for  the  prosecution  of  some  industrial  pursuit. 

4:10.  A   Corporate  Company  or  Corjyoration 

is  a  company  authorized  by  law  to  transact  business  as  an 
individual. 

411.  A  Charter  is  the  legal  document  which  defines  the 
rights  and  obligations  of  a  corporation. 

412.  Capital  Stock  is  the  property  or  money  invested 
in  the  business  of  the  company. 


STOCKS.  269 

413.  A  Share  i-  "'"  "f  tlio  equal  divisions  of  tlio  capital 
stock  of  a  com  pa  II 

The  value  of  a  sliarc  is  difl'trcnt  in  different  companies.  Unless 
otherwise  specified  it  will  be  regarded  as  $100. 

414.  A  Certificate  of  Stock  is  a  paper  issued  by  a 
corporate  company  giving  the  number  of  shares  to  which  the 
holder  is  entitled,  and  the  original  value  of  each. 

415.  l^ar  Value  is  the  value  named  in  the  certificate. 

When  shares  sell  for  the  value  named  on  their  face  the  stock  is  said 
to  be  a/  par;  when  for  more  than  their  face,  above  par,  or  at  a  pre- 
mium;  when  for  less  than  their  face,  below  par,  or  at  a  discount. 

416.  The  Market  Value  is  the  sum  for  which  stocks 
sell. 

417.  An  Installment  is  a  portion  of  the  capital  stock 
paid  by  the  stockholder. 

418.  A  lyividend  is  a  suni  divided  ainoncr  the  stock- 
holders as  the  profits  of  the  business. 

419.  An  AssesMinent  is  a  sum  which  the  stockhold- 
ers of  a  company  are  required  to  pay,  to  make  up  deficiencies 
or  losses. 

The  Government  and  Corporations  frequently  issue  Bonds 
for  the  purpose  of  raising  money. 

420.  A  lion  (I  is  a  written  obligation  under  seal,  securing 
the  payment  of  a  sum  of  money  on  or  before  a  specified  time. 

Interest  is  usually  paid  upon  bonds  at  fixed  dates,  as  an- 
nually or  semi-annually. 

421.  CoupouH  are  certificates  of  interest  attached  to 
bonds.  They  are  cut  off*  and  presented  for  payment  as  often 
as  the  interest  becomes  due. 


270  PERCENTAGE. 

422.  United  States  Grovernment  Securities  are  of  two 
kinds,  viz:  bonds  which  are  to  be  paid  at  a  specified  time, 
and  bonds  which  are  to  be  paid  at  a  fixed  date,  or  some 
earlier  si)ecified  time  at  the  option  of  the  Government. 

Bonds  arc  sometiinefl  designated  by  combining  the  rate  of  interest 
and  the  time  of  payment.  Thus,  bonds  that  pay  0^  interest  and  are 
payable  in  1881,  arc  called  G's  of  '81. 

When  the  rate  is  uniform  for  a  class  of  bonds  it  is  omitted,  and 
the  time  of  re<lemplion  and  payment  only  are  given.  Thus,  bonds 
that  may  be  redeemtHl  in  five  years,  and  are  payable  in  twenty  years, 
are  called  5-20'h,  those  redeemable  in  t.n  v.irs  :mil  i):iv:il»l.-  in  forty 
years  are  called  ICMO's. 

The  4%  consoU  are  the  bonds  of  the  numjuaau'!  i. .m.  i  lu  v  un  re- 
deemable after  30  years  from  July  1,  1877. 

The  bonds  issued  by  States,  counties,  etc.,  are  named  from  the  rate 
of  interest  they  bear.  Tiius,  New  Hampshire  bonds  that  benr  6^ 
interi'st  are  calleil  New  Hampshire  6's. 

All  bonds  of  the  Unit.-.I  f^t:it*'s  are  payable  in  coin  at  maturity. 

The  various  classes  ^/x   I  iiited  States  bonds  are: 

1.  6's  of  '81.     Interest  payable  semi-annually  in  coin. 

2.  5-20's,  issued  in  1862,  '64,  '65,  '67,  '68.  Interest  payable  semi- 
annually in  coin  at  6^. 

3.  10— iO's,  issued  in  1864.  Interest  payable  in  coin  semi-annually, 
at  5%  on  $500  and  $1000  bonds,  and  annually  on  $50  and  $100  bonds. 

4.  5*8    of     '81.     Interest  payable  quarterly  in  coin. 

5.  4J's  of     '86.     Interest .  payable  quarterly  in  coin. 

6.  4'8    of  1901.     Interest  payable  quarterly  in  coin. 

7.  4^    consols.    Interest  payable  quarterly  in  coin. 

423.  Stocks  is  a  term  which  includes  the  stock  of  a 
corporation,  the  various  U.  S.,  State,  and  county  bonds,  etc. 

424.  A  StocL'-broker  is  a  person  whose  business  it  is 
to  buy  and  sell  stocks. 

425.  brokerage  is  the  compensation  allowed  a  broker 
for  buying  and  selling  stocks. 


STOCKS.  271 

426.  The  computations  in  stocks  involve  the  same  elements 
&8  the  fundamental  problems  in  Percentage.  The  correspond- 
ing terms  are: 

1.  The  Par  Value  of  the  stock  is  the  Base, 

2.  The  Rate  of  Fremium,  or  Discount  is  the  Rate, 

3.  The  Premium  or  Discount  is  the  Percentage, 

4.  The  Market  Value  is  the  Amount,  or  Difference, 

427.  Principle. — Brokerage  is  computed  on  the  par  value 
of  the  stock. 

EXERCISES. 

1.  What  is  the  cost  of  SIOOO  Delaware,  Lackawanna, an-l 
Western  R.  R.  bonds  at  108  when  \^  additional  is  paid  for 
buying  ? 

2.'  A  broker  received  82510  to  invest  in  first  mortgage 
bonds  at  a  premium  of  25^.  If  he  charged  ^J^  brokerage, 
how  many  bonds  of  $100  each  did  he  buy? 

3.  A  broker  invested  826000  in  quicksilver  stocks  at 
12|,  chnrprlnpr  1%  brokcnifrc.  How  many  shares  did  he 
buy 

4.  v  inan  i.iM>it(l  ^.'>\2>>  ill  I  hiou  Pacific  R.  R.  bonds 
at  $102,  paying  ^^  brokerage.  How  many  bonds  of  $1000 
each  did  he  buy? 

5.  A  man  bought  a  number  of  shares  of  bank-stock  at 
125  and  sold  it  at  128,  gaining  $300.  How  many  shares 
did  he  buy? 

G.  Bought  bonds  at  llo  and  sold  them  at  110,  lo.«intr 
8300.     How  many  bonds  of  $1000  each  did  I  buy/ 

7.  How  much  must  I  invest  in  Missouri  State  bonds  at 
par,  which  pay  6j^  interest  to  secure  an  income  of  $1200  iin- 
nuully  ? 

8.  What  &uni  must  1  iuvcjit  in  stock  at  par  paying  an 
annual  dividend  of  8^  to  realize  an  income  of  $4800  yearly  ? 


J  < '-!  PERCENTAGE. 

9.  Wlmt  per  cent,  on  the  investment  does  6^  stock  yield 
if  it  is  bought  at  half  its  par  value?    If  at  \  of  its  par  value? 

10.  What  |)er  cent,  does  stock  pay,  if  it  yields  an  annual 
divi<lend  of  4^  and  is  bought  at  50^  of  its  par  value? 

11.  How  much  must  I  pay  a  share  for  stock,  which  yields 
an  annual  dividend  of  6^,  so  that  I  may  rciili/  •  V'^^/  on  my 
money  annually  ?     18^  ?     24  fo  ?     30^  ? 

12.  For  what  price  must  bonds  bearing  9^  interest  be 
bought,  so  that  12^  may  be  realized  annually?     6^  ? 

13.  How  much  currency  can  be  obtained  for  8100  in  gold, 
when  gold  is  at  a  premium  of  4^?     9^  ?     12^  ? 

14.  How  much  currency  can  be  obtained  for  $100  in  gold, 
when  gold  is  at  a  premium*  of  10  fc  ?     100^  ?     150^  ?    5^  ? 

15.  How  much  gold  at  6^  premium  can  I  buy  with  $106 
in  currency?     With  $318?     With  $159? 

16.  When  gold  is  at  a  premium  of  20^ ,  how  much  gold 
is  $1  in  currency  worth.  How  much  when  gold  is  at  50^ 
premium  ? 

17.  When  gold  is  selling  at  125,  what  is  the  value  in  gold 
of  a  United  States  greenback  for  $10? 


WRITTJy     IXERCISES. 

428.    1.  What   is    the  cost  of  500  shares   Delaware  and 
Hudson  Canal  Co.  stock  at  50^,  brokerage  \^  ? 

PROCESS.  Analysis.  —  Si  nee 

501  </  4-  1  C^  —  50»  ^  50^^,  of  the  par  value 

50f^  of  $100  =  $50.75,  cost  of  1  share.      price  paid  for  it,  the 

$50.75x500^-825375,  the  entire  cost.      entire    cost    of    the 

stock,  including  the 
rate  for  brokerage,  is  50^^  of  the  par  value  of  the  stock.  And,  since 
the  par  value  of  a  share  of  the  stock  is  $100,  the  cost  of  a  share  will 
be  50^  ^c  of  $100.  or  $50.75,  and  the  cost  of  500  shares  of  the  stock 
will  therefore  be  500  times  $50.75,  or  $25375. 


<Tni'K<.  273 

Rule. — Since  Vie  ^nm<  ,irnir„L^  a,(  involved  as  in  ihf'fvndn- 
nienial  probleim  in  Percentage,  tlie  niles  are  tlie  same. 

FORMULAS. 

1.  Premium  or  IH8Count=Par  Value  X  Rate. 

2.  Bate  =  Premium  or  Discount  —-  Par  Value. 

3.  Par  Value  =  Premium  or  Discount  -~  Rate. 

'   \{1  —  Rate). 
5.  Market  Value  ^^  Pur  \alue  < 


4.  Par  Value  =  Market  Value  •    ^  ^^  +  ^''^^^' 


Discount. 


2.  Find  the  cost  of  125  shares  Union  Pacific  R.  R.  stock, 
at  68  i,  brokerage  \^  ? 

3.  Wliat  will  $8000  U.  S.  5-20*8,  coupon  bonds  of  65,  cost 
at  108^,  brokerage  ^fc  • 

4.  How  much  will  55  share.s  C   (y.  C  tt  I.   K.  K.  sttx^k 
cost  at  28},  brokerage  ij^  ? 

5.  What  must  be  paid  for  $5000  U.  S.  10-40's.  at  S\fo 
j)remium,  brokerage  1^? 

6.  Bought  35  shares  N.  Y.  C.  &  H.  R.  R.  R.  stock  at 
86 J,  and  sold  them  at  S\^  advance.     How  much  did  I  gain? 

7.  Sold  135  shares  railroad  stock  at  a  discount  of  15^^," 
paying  Jj^  brokerage.     How  much  did  I  receive  for  it? 

8.  How  many  shares  of  bank-stock   at  5^  discount  can 
1)0  purchai?ed  for  $3810,  if  \%  is  jmid  for  brokerage? 

PROCESS.  Analysis.— 

i/\/\^/        r^       i\  ir  ^   ,    t  ^        rtf-1^^  Since  the  stock  wn« 

100%-5^  =  95^4-i%  =95i%         i^^^,^^  ^^  5^^  jj^. 

?  •?  s  1  (»         •»  ->  »        a  t  o  n  (\    .>^   I  0  .hares,      count  it  was  bought 

at   95^  of  its   par 

vaiiif,  1)111  tin-  tjrokcrngf  iiicn-a-i.-d  tiu'  cost  ^r^,  so  that  each  dollar's 

worth  of  Ktiick  cost  95J<^  of  its  par  vahie,  or  S.95}.     Therefore,  an 

iniiny  doUani'  worth  can  be  bonglit  for  $.3810  as  $.95^  is  contained 

tiiijcM  in  $:W10,  which  is  4000  times,  or  40  shares  can  bo  bought 


274  PERCENTAGE. 

9.  Find  the  number  of  shares  of  R,  R.  stock  at  102f,  which 
can  be  bought  for  $2575,  brokerage  |%. 

10.  How  many  shares  of  N.  Y.  C.  &  H.  R.  R.  R.  stock 
at  98|,  can  be  Iwught  for  $28710,  brokerage  i%? 

11.  How  many  shares  of  C.  B.  &  Q.  R.  R.  stock  at  109J, 
can  be  bought  for  $66075,  brokerage  \%? 

12.  How  many  shares  of  Hartlund  Ferry  stock  at  4%  dis- 
count, can  be  bought  for  $3330.25,  bmkemge  J%  ? 

13.  How  many  shares  of  R.  R  stock  at  3%  discount, 
can  be  bought  for  $215a50,  brokerage  J%? 

14.  What  income  will  be  realized  from  investing  $4196.25 
in  5%  stock  purcluised  at  93,  allowing  \%  for  brokerage? 

PROCESS.  Analysis. — 

$4196.25-s-.93i  =  $4500,theparvaluo.   f^gjjj"^^'; 

$4500  X  •05  =  $22  5,  annual  income.  its  par  value, 

every  dollar's 
worth  coBt  $.93^;  and  as  many  dollars'  worth  can  be  bought  for 
$4106.25  as  $.93}  is  contalne<l  times  in  that  sum,  which  is  4500 
times;  and  since  the  stock  paid  b^c  income,  the  entire  income  from 
$4500  Is  5^  of  $4500,  which  is  $225. 

15.  How  much  income  will  I  receive  annually  by  investing 
$1299  in  6%  stock  purchased  at  37%,  allowing  \%  bro- 
kerage? 

16.  What  will  be  the  income  from  investing  $4696.25  in 
Crawford  Co.  6's  at  45,  brokerage  \%. 

17.  Which  is  more  profitiible,  and  how  much,  to  invest 
$5000  in  6^  stock  purchased  at  75 5^,  or  5%  stock  purchased 
at  60%? 

18.  U.  S.  5-20's  pay  6%  interest  in  gold.  What  will  be 
my  income  in  currency  by  investing  $11212.50  at  112|^,  when 
gold  is  quoted  at  6J%  premium? 

19.  Which  is  more  profitable,  to  buy  6%  bonds,  purchased 
at  90,  interest  payable  in  currency;  or  5%  bonds,  purchased 


.siM<  iv.-^.  275 

at  95,  interest  payal)le  in  gold,  when  gold  is  quoted  at  6J% 
premium?  How  much  more  profitable  in  currency  is  it  on 
each  $100  invested  ? 

20.  A.  B.  Howard  sold  a  mill  lur  $13850,  which  had  been 
paying  an  annual  profit  of  5J^  of  that  sum,  and  invested 
the  proceeds  in  U.  S.  10-40's  at  lllf,  paying  ^^  brokerage. 
Was  his  yearly  income  increased  or  diminished,  and  how 
much  in  currency,  gold  being  at  5%  premium? 

21.  How  much  must  be  invested  in  6%  stock,  purchased 
at  90,  to  secure  to  the  purchaser  an  income  of  $900  annually? 

PROCESS.  Analysis. — 

$900--.06  =  $15000,  par  value  of  stock.  ^^^"^  of^lls 
$15000X.90  =  $13500,  cost  of  stock.  $.00,    it    will 

require  as 
many  dollars  to  secure  an  income  of  $900  as  $  .06  is  contained  times 
in  $900,  which  is  15000  times;  and  since  the  stock  is  selling  at  ^O^o 
of  its  par  value,  90^  of  $16000,  which  is  $13500,  will  be  the  cost. 

22.  I  desire  to  invest  in  Ohio  &  Mississippi  R.  R.  Iwnds 
which  bear  6%  interest,  a  sum  of  money  sufficient  to  bring 
an  income  of  $1000.  If  the  bonds  can  be  bought  at  91%, 
how  much  money  must  I  invest,  brokerage  ^^  ? 

23.  What  sum  must  I  invest  in  Louisiana  7's  at  107J  to 
secure  an  annual  income  of  $1750? 

24.  What  sum  must  I  invest  in  U.  S.  6's  at  112|  to  se- 
cure an  annual  income  of  $1750? 

25.  What  sum  must  be  invested  in  6%  stock  at  $84.50 
per  share  to  yield  an  income  of  $900  annually? 

26.  What  per  cent,  income  on  my  investment  will  I  re- 
ceive if  I  buy  6%  stock  at  20%  premium? 

PROCESS.  Analysis. — Since  $1    of 

1 . 0  6  -5-  $  1 .  20  = .  0  5,  or  5%.       ^''^  J"^^^  <^^«  ^^•''^'  *"*^ 

the  income  from  it  is  $.06, 
the  income  is  jl^,  or  ,V,  or  5^  of  the  amount  of  the  investment. 


276  PERCENTAGE. 

27.  What  18  the  rate  per  cent,  of  income  from  bonds  wliicL^ 
pay  7%  interest  when  they  are  bought  at  105? 

28.  If  stock  which  pays  a  semi-iinnual  dividend  of  o\% 
be  Iwught  at  10%  premium,  what  rate  per  cent,  of  inconm 
does  it  pay? 

29.  AVIiich  affords  the  greater  per  cent,  of  income,  bonds 
b:)Ught  at  125  which  jmy  8%,  or  bonds  which  pjiy  (')%  bought 
at  a  discount  of  10^  ? 

30.  Which  is  more  profitable,  and  how  much  per  cent.,  lo 
buy  New  York  7's  at  105^,  or  Louisiana  6's  at  98^  ? 

-  31.  What  per  cent,  of  income  does  stock  paying  9%  divi- 
dends afford  if  it  is  bought  at  112? 

.    32.  How  much  must  I  pay  for  New  York  6*s  so  that  I 
may  realize  an  income  of  9^  on  the  investment? 

PROCESS.  Analysis. — Since  tlic  income  is  C^ 

^  ^  06  -T-  $  .  09  =  .  66-8-         ^^  every  dollar  of  the  par  value  of  the 

Block,  if  an  income  of  9^  on  an  invest- 
ment be  desired,  then  6^  of  the  par  value  of  the  stock  must  be  9^  of 
the  sum  paid  for  $1  of  the  stock,  which  is  $  .663,  or,  the  stock  must 
be  bought  for  66§^  of  its  par  value. 

33.  How  much  must  I  pay  for  stock  wliich  pays  a  divi- 
dend of  15^  so  that  I  may  realize  7^  on  my  investment? 

34.  How  much  premium  must  I  pay  on  stock  which  pays 
a  10%  dividend  so  that  I  may  realize  8^  on  my  investment? 

35.  At  what  price  must  I  buy  7%  stocks  so  that  they  may 
yield  an  income  equivalent  to  10%  stocks  at  par? 

36.  What  must  I  pay  for  New  York  6's  so  that  my  pur- 
chase may  yield  me  7%  ? 

37.  At  what  price  must  I  purchase  15^  stock  that  it  may 
yield  the  same  income  as  6^  stock  purchased  at  90? 

38.  What  is  the  currency  value  of  -^9280  in  gold,  Mhen 
gold  is  selling  at  107|? 

39.  AVhat  is  the  currency  value  of  §7225  in  gold,  when 
gold  is  at  a  premium  of  S^^  ? 


INSURANCE.  277 

40.  What  is  the  value  iu  gold  of  $5000  in  ciirreucy,  when 
the  premium  on  gold  is  6^^? 

41.  When  gold  is  selling  at  105:J^,  what  is  the  value  of 
$7250  in  currency? 

42.  The  net  earnings  of  a  company  whose  capital  stock  is 
$2000000  was  $135000.  If  they  reserve  $5000  as  a  surplus 
fund,  what  per  cent,  dividend  '^an  they  declare? 

43.  Mr.  A.  purchased  250  shares  of  stock  at  75,  in  a 
company  whose  cjipital  was  $1500000.  The  gross  earnings 
of  the  company  for  1876  were  $225000,  the  expenses  were 
40^  of  tiie  gross  earnings,  and  they  reserved  a  surplus  fund 
of  $10000.  What  per  cent,  dividend  did  Mr.  A.  receive  on 
his  investment? 

44.  A  capitalist  owning  200  shares  of  stock  of  $150  \)vv 
share,  on  which  he  was  receiving  a  dividend  of  3%  senii- 
anni/ally,  exchanged  them  for  6^  bonds  purchased  at  98^ 
Did  he  gain  or  lose,  and  how  much  anrually? 

45.  If  a  man  who  had  $5000  of  U.  S.  6's  of  '81  should  sell 
them  at  115,  and  invest  in  U.  S.  10-40's  purchased  at  105, 
would  he  gain  or  lose,  and  how  much  annually? 


INSURANCE, 

420,  1.  How  much  must  I  pay  to  secure  myself  against 
loss  by  fire,  or,  inmre  my  property  for  $5000,  if  an  annual 
sum  or  premium  of  Ij^  is  charged  by  those  who  take  the  risk? 

2.  What  will  be  the  annual  premium  for  insuring  property 
for  $10000  at  ij^  ? 

3.  What  will  be  the  annual  premium  for  insuring  property- 
for  $1500  at  l^^^^ 

4.  What  is  the  premium  at  }^  for  insuring  a  vessel  and 
cargo  to  the  amount  of  $36000? 

5.  What  will  it  rn9t  to  injure      .  ^^  nli  n\'  t.  a  at   H^  ? 


278  PERCENTAGE. 

6.  What  must  be  paid  for  insiiriug  a  building  valued  at 
13000,  for  }  of  its  value  at  l{fo  ? 

7.  If  a  merchant  insured  his  goods  for  $2000  at  2^ ,  how 
much  premium  did  he  pay? 

8.  How  much  premium  did  a  merchant  \y&y  who  insured 
his  stock  of  boots  and  shoes  for  86000  at  1\^  ? 

9.  A  man  paid  $25  premium  for  insuring  his  house  and 
furniture  against  loss  or  damage  by  fire.  For  how  much  was 
he  insured,  if  tlie  rate  of  insurance  was  \^  ? 

10.  For  how  much  was  a  man  insured  who  |>aid  650  pre- 
mium for  insuring  his  barn  and  live  stock  at  1^  ? 

11.  How  much  was  the  amount  of  insurance  wlien  the 
premium  piid  for  insuring  a  house  and  furniture  at  |^  was 
$75? 

12.  For  how  mucli  is  a  larmer  insured  on  his  barns  and  tlie 
grain  in  them,  who  pays  $60  premium,  when  the  rate  is  2^? 


DEFINITIONS. 

430.  Jus u ranee  is  indemnity  against  loss  or  damage. 
It  is  of  two  kinds :  Property  Insurance  and  Personal  Insurance, 

CASE  I. 
PROPERTY  INSURANCE. 

431.  Property  Insuranee  is  indemnity  against  loss 
or  damage  by  fire,  or  Fire  Insurance;  against  loss  or  damage 
by  casualties  at  sea,  or  Marine  Insurance;  and  against  loss  or 
damage  by  fire,  lightning,  etc.,  to  cattle,  horses,  etc.,  or  lAve 
Stock  Insurance, 

432.  The  I^olicy  is  the  contract  or  agreement  between 
the  insurance  company  and  the  person  insured. 

433.  The  Premium  is  the  sum  paid  for  insurance. 


INSURANCE.  279 

The  insuniiice  business  is  conducted  mostly  by  corporate 
comi)anies,  which  are  either  Mutual  or  Stock  companies. 

434.  A  Mutual  Insurance  Coniiyany  is  one  in 

which  the  person  insured  participates  in  the  profits  and  shares 
the  losses  of  the  company. 

435.  A  Stock  Insurance  Company  is  one  in 

which  the  capital  is  contributed  by  stockholders,  who  alone 
participate  in  the  profits  and  share  the  losses. 

A  few  companies  combine  tlie  features  of  both  stock  and  mutual 
companies.    They  are  called  Mixed  Companies. 

436.  The  computations  in  insurance  involve  the  same  ele- 
ments as  do  the  fundamental  problems  in  Percentage.  The 
corresponding  terms  are: 

1.  The  Amount  Luured  is  the  Hasc, 

2.  The  Kate  of  Premium  is  the  Mate, 

3.  The  Premium  is  the  Percentage, 

WRITTEN     EXERCISES. 

1.  How  much  is  the  premium  on  a  policy  of  insurance  on 
a  dwelling  for  33000  if  the  rate  of  premium  is  1^%  ? 

PROCESS.  Analysis. — Since  the  pre- 

$3000  X  .01i  =  $37.*50         "'•""^  "  H/^^  of  the  amount 

insured,  to  find  the  premium, 
\\1lo  of  $3000  must  be  found,  which  is  $37.50. 

Rule. — Since  the  same  elements  are  involved  im  in  the  Junda- 
menial  problems  in  Perceniage  the  ndes  are  the  same. 

FORMULAS. 

1.  Sum  insured  X  Rate  =  Premium. 
J.  Premium  -:  Sum  insured  =-  Rate. 
;>.    I*reiniinn    :    I^te  -    Siini   insured. 


280  PERCENTAGE. 

2.  How  much  is  the  premium  for  insuring  a  stock  of  goods 
for  $15000  at  1J%  ? 

3.  A  man  had  his  house  insured  for  $5000  paying  }^, 
and  his  furniture  for  $3000  paying  1^.  How  much  was 
the  premium? 

4.  How  much  must  be  paid  for  insuring  a  flouring  mill 
valued  at  $18000  for  §  of  its  value  at  2J^  ? 

5.  A  vessel  valued  at  $80000,  with  its  cargo  valued  at 
$65000,  was  insured  for  }  of  its  value  at  IJ^.  What  was 
the  premium?  What  would  be  the  actual  loss  to  the  insur- 
ance  company  if  the  above  vessel  should  be  lost  at  sea? 

6.  Property  was  insured  for  $15850  at  S\fc .  What  amount 
.of  premium  was  jmid  ? 

7.  Paid  $275  for  insuring  property  at  J^.  What  was  the 
amount  insured? 

8.  Paid  $325  for  insuring  property  valued  at  $16250. 
What  w^as  the  rate  of  premium  ? 

9.  A  insured  his  buildings  for  $9oOO,  paying  a  premium 
of  $47.50.     What  rate  did  he  pay? 

10.  Mr.  Orcott  paid  $1:75  for  insuring  his  block  of  mer- 
cantile buildings  at  1\^.  How  much  insurance  did  he 
procure  ? 

11.  Mr.  James  paid  $652.50  for  insurmg  property  valued 
at  $43500.     What  was  the  rate? 

12.  A  man  paid  $180  for  insuring  his  saw  mill  for  |  its 
value  at  3^.     What  was  the  value  of  the  mill? 

13.  A  merchant  whose  stock  of  goods  was  valued  at  $30000 
insured  it  for  three-fourths  of  its  value  at  J^.  In  a  fire  he 
saved  $5000  of  the  goods.  What  was  his  loss?  What  was 
the  loss  of  the  insurance  companies? 

14.  The  price  of  a  quantity  of  silks  was  discovered  by 
knowing  that  they  were  insured  at  4J^  for  two-thirds  of 
their  value,  and  the  premium  paid  was  $400.  What  was  the 
price  of  the  silks? 


INSURANCE.  281 

15.  A  manufacturing  company  insured  its  works  for  |  of 
their  value  at  2^^,  paying  as  premium  $1657.50.  What 
was  the  value  of  the  works? 

16.  A  vessel  and  c^rgo  were  insured  for  -f  of  their  value 
at  1J%.  Tiie  premilini  amounted  to  82475.  At  what  price 
were  the  vessel  and  cargo  valued? 

17.  Paid  $225  for  insuring  a  store  and  its  contents  for  f  of 
their  value  at  1^^.  The  stock  was  worth  half  as  much  as 
the  store.     What  was  the  value  of  each? 

18.  For  how  much  must  a  block  of  stores  worth  $20000 
be  insured,  so  that  the  insurance  will  cover  J  of  the  value 
of  the  property  and  the  amount  of  the  premium  at  1^%? 


CASE  II. 
PERSONAL  INSURANCE. 

437.  Personal  Insurance  is  indemnity  against  loss 
of  life,  or  Life  Lisurance;  against  loss  occasioned  by  accidents, 
or  Acxndent  Lisurance;  and  against  loss  occasioned  by  sickness, 
or  Health  Inmrancr. 

The  policies  issued  by  Life  Insurance  Companies  are  of 
various  kinds,  the  chief  of  which  are  the  Life  Policy  and  the 
Endowment  Policy. 

438.  A  Life  Policy  secures  a  sum  of  money  at  the 

death  of  the  |)erson  insured. 

430.  An  Endonrment  Policy  secures  a  sum  of 
money  at  a  specified  time,  or  at  death,  if  it  occur  before  the 
8|)ecified  time. 

440.  Accident  and  Ucdlth  Policies  secure  ;i  .stij) 
ulated  sum  for  a  certain  time,  in  case  of  a  disabling  accident 
or  sickness,  and  the  face  of  the  policy  in  case  of  death  by 

accident. 


282  PERCENTAGE. 

441.  The  rates  of  premium  are  based  upon  the  expedoHon 
of  life,  determined  by  observing  the  death  rate  per  thousand 
ijihabitants. 

]\  i:  I  T  TEN    EXERCISES. 

442.  1.  What  premium  must  be  paid  annually  for  a  life 
policy  of  $5000,  at  $21.10  per  $1000? 

Analysis.— Since  the  rate  of 

PROCESS.  premium  i«  $21.10  annually  for 

$21.10X6  =  $105.50       $1000,  the  premium  for  $5000  is 

5  times  $21.10,  which  is  $105.50. 

2.  How  much  will  be  the  annual  premium  on  a  life  insur- 
ance policy  for  $3000  at  $31.30  per  $1000? 

3.  What  is  the  amount  of  annual  premium  on  a  life  policy 
for  $5500  at  $26.38  per  $1000? 

^  4.  If  a  person  who  is  insured  for  $5000,  at  an  annual  pre- 
mium of  S28.90  per  $1000,  dies  after  9  payments,  how  much 
more  will  his  heirs  get  than  has  been  paid  in  premiums? 

5.  If  a  man  insures  his  life  for  $5000,  paying  $22.90  per 
$1000,  and  dies  immediately  after  paying  his  annual  premium 
for  30  years,  what  is  the  result  of  the  investment,  reckoning 
simple  interest  at  6%  on  the  premiums  paid? 

6.  If  Mr.  Bowditch  insures  his  life  for  $5000  on  the  endow- 
ment plan  when  he  is  40  years  of  age,  the  policy  to  be  payable 
when  he  is  55  years  of  age,  paying  therefor  $54.90  for  $1000, 
will  he  gain  or  lose  by  insuring,  reckoning  simple  interest  at 
7%  on  the  premiums  paid,  if  he  lives  till  the  policy  is  paid? 

7.  A  traveling  agent  has  an  accident  insurance  policy  for 
$3000,  for  which  he  pays  $50  per  year.  His  weekly  com- 
pensation, in  case  of  a  disabling  injury,  is  $30.  Immediately 
after  he  makes  his  fifteenth  payment  he  is  disabled  by  an  in- 
jury for  20  weeks.  Does  he  gain  or  lose  by  the  insurance, 
reckoning  simple  interest  at  Q%  on  the  premiums  paid? 


-'*% 


yEmc^AHGT 


443.  1.  When  A  owes  B  $500,  and  B  owes  A  3500,  how 
may  the  accounts  be  settled  without  any  transfer  of  money 
taking  place? 

2.  When  A  in  Chicago  owes  B  in  New  York  8500,  and  C 
in  New  York  owes  A  81000,  how  can  A  pay  his  indebted- 
ness to  B  without  remitting  the  money? 

3.  What  will  be  the  indebtedness  of  A,  B  and  C  to  each 
other  after  the  transaction  has  taken  place? 

4.  A  and  C  live  in  the  same  city,  and  B  in  a  distant  city. 
A  owes  B  32000,  and  B  owes  C  31000.  How  may  B  pay 
his  indebtedness  to  C  without  remitting  the  money? 

5.  What  will  be  their  indebtedness  to  each  other  after  A 
has  imid  B's  order,  or  draft  f 

6.  What  will  a  draft  for  3500  cost,  payable  when  it  is 
j)resented,  or  at  sight,  if  ^%  premium  is  charged  for  it? 

7.  How  much  should  be  deducted  from  the  price  of  the 
above  dmft  if  it  is  not  to  be  paid  until  two  montlis  money 
iKjing  worth  6%  ? 

8.  What  will  l)e  the  cost  of  a  dralt  lor  ?5oU,  payable  at 
sight,  if  it  is  purchased  at  1%  discount? 

9.  What  will  be  the  cost  of  a  sight  draft  for  3300,  pur- 
chased at  J%  premium? 

10.  If  A  in  Nashville  owes  B  in  New  Orleans  31000,  and 
C  in  New  Orleans  owes  I)  in  Na.«hville  31500,  how  may  A 
jwiy  his  indebtedness  without  n'mittinir  thr  money? 

'   (2K3) 


284  PERCENTAGE, 

11.  If  the  premium  is  J^,  how  much  will  it  cost  me  to 
remit  a  draft  for  8800  from  Cincinnati  to  Cleveland? 

12.  If  a  man  sells  a  draft  for  $500,  at  a  premium  of  j^, 
how  much  does  he  receive  for  it? 

13.  A  wishes  to  send  to  his  agent  in  New  Orleans  a  draft 
for  85000.  If  the  premium  on  exchange  is  f^,'how  much 
will  the  dnift' cost, him?  .  - 

14.  When  I  pay  82025  for  a  sight  draft  on  New  York  for 
82000,  what  is  the  premium,  or  rat^  of  exchange? 

15.  When  I  can  buy  a  sight  draft  on  Chicago  for  82000, 
paying  for  it  81980,  what  is  the  rate  of  exchange? 

16.  If  Mr.  Burt  pays  84975  for  a  sight  draft  on  Cincinnati 
for  85000,  at  what  rate  is  exchange? 


DEFINITIONS. 

444.  Exchange  is  the  method  of  making  payments  in 
distant  places  without  transmitting  money. 

44o.  A  Draft  or  Bill  of  Exchange  is  a  written 
order  by  one  person  to  another  to  pay  a  specified  sum  of 
money  to  the  person  named  in  the  writing  or  his  order. 

FORM   OF   A    DRAFT. 

$384^^.  Cincinnati,  O.,  July  20, 1877. 

Tiventy  days  after  sight  pay  to  the  order  of  Hie  First  Na- 
tional Banky  Chicago,  III.,  Three  Hundred  Eighty -four  3% 
Dollars,  value  received,  and  charge  to  the  account  of 

To  James  H.  Hoose  &  Co.,  J^^ES  Bros.  &  Co. 

Chicago,  111. 

There  are  primarily  three  parties  connected  with  a  draft,  viz : 
the  person  who  signs  it,  the  person  who  is  ordered  to  pay  the 
money,  and  the  person  to  whom  the  money  is  to  be  paid. 


KXLHA-NGK.  285 

446.  The  jyrawer  is  the  person  who  sigm  the  draft. 

447.  The  Drawee  \?  tho  person  who  is  ordered  to  pay 
the  money. 

448.  The  Payee  is  the  person  to  whom  it  is  ordered 
that  the  money  be  paid. 

449.  A  Sight  Uraft  is  one  which  is  to  be  paid  when 
it  is  presented  to  the  drawee. 

450.  A  Time  Draft  is  one  payable  at  a  specified  time, 
after  its  presentation  to  the  drawee,  or  after  date. 

On  Time  Drafts  three  days  of  grace  are  usually  allowed. 

451.  Accepting  a  draft  is  agreeing  to  pay  it  when  it 
is  due.  This  is  done  by  the  drawee  writing  "Accepted" 
acrojss  the  face  of  the  draft,  with  his  name  and  the  date. 

Exchange  is  of  two  kinds,  viz :  Domestic  or  Inland^ 
and  1^0  reign, 

CASE  I. 
DOMESTIC   EXCHANGE. 

452.  Domestic  Exchange  treats  of  drafts  payable 
in  the  country  where  they  are  made. 

irniTTEN    EXEBCISES, 

453.  1.  What  will  be  the  cost  of  procuring  a  sight  draft 
on  New  York  for  $5000  at  i%  premium? 

PROCESS.  Analysis. — Since  exchange 

«14-«   OOJl-1   004  *""  ^''''''  ^""'^  '"  "*  -^  ^''" 

f  1  -h  » .  u u  ^  —  1 .  u  u  *  ^.^^^^^  ^^.^j.^,  5j  ^f  j,^^^  j^^^f^ 

$1.00iX5000r=$5025       will  cost  $1.00 \,   and  a  draft 

for  $5000  will   therefore  cost 
the  purchaser  5000  times  $1.00|,  which  is  $5025. 


286  PERCENTAGE. 

2.  What  will  be  the  cost  in  New  Orleans  of  a  draft  on 
New  York,  payable  60  days  after  sight,  for  $5000,  exchange 
being  at  1^%  premium?  AKALvs«.-^Si„c« ,he 

PROCESS.  exchange  on  New  York 

$1.  +  ».016=«1.015  '■'  "'^M  P""!""' 

'  every  $1   of   the   draft 

$1.015  — |.008}  =  S1.006J  would    cost   $1,015    if 

$1,006^X5000  =  15031.25        paidatsight.    Butsince 

it  is  not  to  be  paid  in 
New  York  for  63  days,  the  banker  in  New  Orleans  who  haa  the  use 
of  the  money  for  that  time,  since  he  is  not  obliged  to  pay  the  money 
in  New  York  for  03  days,  allows  the  bank  discount  on  the  face  of  the 
draft  for  that  time.  The  bank  discount  on  $1,  at  the  legal  rate  in  the 
State  of  Louisiana,  for  the  given  time,  is  $.008),  which,  subtracted 
from  $1,015,  glvcB  $1.0064,  the  cost  of  $1  of  the  draft.  Since  the  cost 
of  a  draft  for  $1  is  $1,006^,  the  cost  of  a  draft  of  $5000  will  be  5000 
times  $1.0061,  or  $5031.25. 

3.  What  will  be  the  cost  in  Memphis,  Tenn.,  of  a  sight 
draft  on  Cincinnati  for  $1000,  the  rate  of  exchange  being 
^^   premium? 

4.  If  exchange  on  Chicago  is  IJ^  premium,  what  will  be 
the  cost  in  Savannah,  Ga.,  of  a  sight  draft  for  S3000? 

5.  A  merchant  in  Chicago  bought  a  draft  on  New  York 
for  $5000,  payable  1  mo.  after  sight.  What  did  it  cost  if 
exchange  was  1%  premium? 

().  What  will  be  the  cost  in  Detroit,  Mich.,  of  a  draft  for 
$1500  on  Cleveland,  O.,  j^ayable  3  mo.  after  date,  wlien 
exchange  is  \%  discount? 

7.  How  much  will  be  realized  from  the  sale  of  a  draft  for 
SoOOO  at  J%  discount? 

8.  How  much  will  be  realized  from  the  sale  of  a  draft  for 
83000  sold  at  J-%  premium  ? 

9.  When  exchange  is  at  J%  premium,  what  will  be  the 
cost  of  a  draft  for  $5000,  purchased  in  Cliicago  on  Omaha, 
to  be  paid  3  mo.  after  date? 


KXCIIANGE.  287 

10.  If  exchange  is  at  \%  premium,  what  will  a  draft  for 
$1500  cost,  purchased  in  St.  Paul,  Minn.,  on  Dayton,  O., 
payable  in  2  mo.,  without  grace? 

11.  If  exchange  is  at  ^f^  premium,  what  will  a  draft  for 
85000  on  New  York  cost  in  Cincinnati,  payable  2  mo.  after 
date? 

12.  How  large  a  draft  on  New  Orleans  can  be  purchased 
for  85000,  when  exchange  is  at  1^^  premium? 

PROCESS.  Analysis. — Since  ex- 

$5000-^-1.015  =  84926.11        tobuy  a  draft  for$l,  and 

$5000  will  buy  a  draft  for 
as  many  dollars  an  $1,015  is  contained  times  in  $5000,  or  $4926.11. 

13.  How  large  a  draft  ov  Washington,  D.  C,  payable 
60  days  after  sight,  can  be  bought  in  Nashville,  Tenn.,  for 
83000,  when  exchange  is  at  1^  discount? 

PROCESS.  Analysis. — Since    ex- 

ftl        «ni_ftQQ  change  is  at  1^  discount, 

^>1  — «.Ul_^.yy  it  would  cost  $.99  to  buy 

$.99  —  8.0105  =  8.9795  a  draft  for  $1   if  it  were 

83000-4-. 0795  =  83062. 78       payable  at  sight ;  butsince 

the  draft  is  not  to  be  paid 
until  63  days,  the  banker  in  Nashville  who  has  the  use  of  the  money 
for  63  days,  allown  !)ank  discount  on  the  face  of  the  draft  for  that 
time,  or  $.0105  for  every  dollar.  Therefore  since  it  costs  $.9795  to 
purchase  a  draft  for  $1,  $3000  will  purchase  a  draft  for  as  many 
dollars  as  $  .9795  is  contained  times  in  $3000,  or  $3062.78. 

1 4.  How  large  a  sight  draft  can  be  purchased  on  Chicago 
for  85725,  when  the  rate  of  exchange  is  |^  premium? 

15.  AVhat  will  be  the  face  of  a  30-day  draft  pundiased  for 
81500,  if  the  rate  of  exchange  is  i^  premium  and  the  rate 
of  discount  is  6%  ? 

10.  If  I  pay  81200  for  a  dnift  payable  in  2  mo.,  when 


2SS  i»p:rcentage. 

the  premium  on  exchange  ia  |^,  and  the  rate  of  discount 
16  9^ ,  what  will  be  the  fiM»  of  the  draft  ? 

17.  How  large  a  sight  draft  on  New  York  can  be  pur- 
chased in  Chicago  for  $10000,  if  exchange  is  i^  discount? 

18.  A  commission  merchant  in  St.  Louis,  Mo.,  sold  goods 
amounting  to  $3500  for  a  man  in  Denver,  Col.  He  sent 
tlie  amount  due  by  a  draft  payable  in  1  nu).  after  sight,  ex- 
change being  \^  premium.  How  large  a  draft  did  he  pur- 
chase? 

19.  How  large  a  draft  at  sight  on  San  Francisco  can  I 
purchase  for  $1750  if  exchange  is  at  ^^  premium? 


C.\SE  II. 

FOREIGN   EXCHANGE. 

454.  Jt^orcif/n  JRxchattf/e  treats  of  drafts  made  in 
one  country  and  payable  in  another. 

Foreign  bills  of  exchange  in  the  United  States  are  drawn  on  London, 
Paris,  Berlin,  Antwerp,  Amsterdam,  Hamburg,  Bremen,  and  other  com- 
mercial centers;  but  drafts  on  London  and  Paris  are  more  common, 
inasmuch  as  they  are  paid  anywhere  on  the  continent  of  Europe. 

455.  A  Set  of  Exchnufje  consists  of  three  drafts  or 
bills  of  the  same  date  and  tenor,  named,  respectively,  the 
first,  second,  and  third  of  exchange.  They  are  sent  by  differ- 
ent mails,  so  that  if  one  is  lost  another  may  be  presented. 
When  one  bill  of  the  set  is  paid  the  others  ai'e  void. 

456.  The  value  of  a  pound  sterling  previous  to  1873  was 
fixed  at  S4.44f  In  1873  Congress  fixed  the  value  of  the 
sovereign  in  U.  S.  gold  coin  at  $4.8665,  which  is  now  the  par 
of  exchange. 

The  value  of  a  franc  is  about  $  .193,  or  about  5.18  francs 
to  one  dollar  in  gold.  Exchange  on  Paris  is  quoted  at  a  cer- 
tain number  of  francs  to  a  dollar  in  gold. 


r..\.  ii.\N(.i:.  289 


H   /;  /  /   /   /    V     IX'ERCISES. 


457.  1.  What  is  the  cost  in  currency,  in  New  York,  of  a 
siirlit draft  on  Lr>n(lon  for  £312  15s.  od.,  ^vhen  exchange  is 
S4.<H7  for  a  pDiind  sterling  and  gold  at  lOG. 

ANALYSIS. 

£312  159.  5d.  =:  £312.7708,  value  in  pounds  and  decimals  of  a  pound. 
$1.87  X  312.7708  =  $152:1193  +,  the  cost  in  U.  S.  gold. 
$1523.193  X  1.06  =  $1614.59,  the  cost  in  U.  S.  currency. 

2.  How  large  a  bill  of  exchange  at  sight  on  London  can 
l)e  bought  in  New  York  for  $2984.38  in  currency,  exchange 
being  at  $4.86  for  a  iK)und  sterling  and  gold  at  107^? 

ANALYSIS. 

$2984 .38-!- $1.07 ^  =  $2776.1 67,  value  of  currency  in  gold. 
$2716.167 -T- $4.86=  £571.2263 -{-,  value  in  pounds  sterling. 
£57 1.2263  =  £571,48.  6|d.,  the  face  of  the  draft. 

3.  How  large  a  bill  of  exchange  at  sight  on  London  can 
l)c  l)ought  in  New  York  for  $3762.50  in  currency,  when  gold 
is  at  lOoi  and  exchange  is  at  $4.87? 

4.  What  will  be  the  face  of  a  sight  dmft  on  London, 
which  is  purchased  in  Philadelphia  for  $5928.75  in  currency, 
when  gold  is  at  106^^  and  exchange  is  at  $4.85^? 

5.  What  will  be  the  face  of  a  sight  drafl  on  London, 
which  is  purchased  in  Norfolk,  Va.,  for  $5575.20  in  cur- 
rency, when  exchange  is  at  $4.87^,  and  gold  is  selling  at 
107}?. 

6.  What  must  be  paid  in  currency  for  a  bill  of  exchange 
on  Paris,  at  sight,  for  3269  francs,  exchange  being  at  5.15 
francs  to  the  dt)llar,  and  gold  at  105|? 

7.  What  must  be  paid  in  currency  for  a  sight  bill  of  ex- 
change on  Paris  for  8950  fnincs,  exchange  being  at  5.19 
francs  for  one  dollar,  and  gold  at  106|? 

19 


290  AVKRAGE   OF   PAYMENTS. 

8.  What  win  be  tlie  face  of  a  sight  draft  on  Paris  which  is 
bought  in  Baltimore  for  $1575  in  curffency,  when  the  rate  of 
exchange  is  5.19  francs  for  a  dolhir,  and  gold  is  at  107;^^? 

9.  An  American  l)ought  a  sight  draft  on  Paris  for  5725 
francs.  AV^hat  waa  the  currency  value  of  the  draft  when  ex- 
change was  at  5.20  francs  for  a  dollar  and  gold  was  106J^? 

10.  What  is  the  value  in  currency  of  a  bill  of  exchange, 
at  sight,  on  London,  for  £895  10s.,  when  exchange  is  $4.87 
f()r  a  iK)uud  sterling  and  gold  is  quoted  at  106|? 

IL  An  American  in  Pliiladelphia  purchased  a  sight  draft 
on  London  for  £585  lOs.  5d.  What  was  the  currency  value 
of  the  draft,  if  exchange  was  at  jwir  and  gold  at  107^? 

12.  What  will  be  the  cost,  in  currency,  of  a  sight  bill  of 
exchange  on  London  for  £875  5s.  4d.,  when  exchange  is 
$4.88^  for  a  pound  sterling  and  gold  is  quoted  at  104^? 


AVERAGE  OF  PAYMENTS. 

458.    1.  How  long  may  $1  be  kept  to  balance  the  use  of 
$5  for  2  months? 

2.  How  long  may  $1  be  kept  to  balance  the  use  of  $7  for 

3  months? 

3.  How  long  may  810  be  kept  to  balance  the  use  of  $5  for    ^ 

2  months? 

4.  How  long  may  $20  be  kept  to  balance  the  use  of  $10  for 

4  months? 

5.  How  long  may  $50  be  kept  to  balance  the  use  of  $25  for 
7  months? 

6.  How  long  may  $40  be  kept  to  balance  the  use  of  $80  for 

3  months? 

»7.  I  owe  B  two  debts  of  equal  amount,  one  due  in  3 
months  and  the  other  in  6  months.  When  may  I  pay  both 
at  one  payment? 


AVERAGE   OF    PAYMENTS.  291 

8.  I  owe  A  two  debts  of  S20  each,  one  due  in  2  months 
and  the  othrr  in  4  months.  When  may  I  pay  both  at  one 
payment 

9.  If*  1  |>:iy  c^i>  ihree  months  before  it  is  diu-,  how  lung 
after  it  is  due  may  I  keep  $30  to  bahmce  it? 

10.  If  I  owe  820,  due  in  4  month.s,  and  S40,  due  in  6 
months,  n»  ul' '♦  tlni.>  ran  I  equitably  pay  both  debts  at  one 
payment . 

11.  If  1  owe  S.'Jl)  due  in  3  months,  and  $10  due  in  7 
months,  whcii  nmy  I  equitably  pay  both  debts  by  a  single 
payment  of  $40? 

DEFINITIONS. 

450.  Averaf/inf/  Paymentft  is  finding  .the  equitable 
time  for  discharghig,  by  one  payment,  sums  due  at  diilerent 
times. 

400.  The  Average  Time  is  the  date  at  which  the  debts 
may  be  equitably  discharged  by  a  single  payment. 

461.  The  Term  of  Credit  is  the  time  that  must  elapse 
before  the  debt  becomes  due. 

462.  Tlie  Average  Term  of  Credit  is  the  time  that 
must  elapse  before  the  debts  due  at  different  times  may  be 
equitably  discharged  by  a  single  jmyment. 


CASE  I. 

463.  Whon    the  terniN  ot*  credit   bo^^in  nf  tlio  Krtino 
dat4>. 

1.     A.    T.    SlvWUrt    it    Co.    M»hl   II    bill    ul    jztMMlN    iipdil    liu-    inl- 

lowing  term.s:  S400  ca.sh,  $300  due  in  2  months,  and  $400 
due  in  4  months.  At  what  time  might  the  whole  indebted- 
ness 1)0  efpiitnblv  ilisiliMi'"-,'.]  hv  :i  K-\\<\\  n.'iviiunt  *' 


292  AVi:iiA(.iE  uF   1' \^  Nt  i;Ni>. 

PROCESS.  Analysis. — Since 

$400  for    0  mo.  =  $1  for  .  f*^  "^^^  *?  ^  P^^** 

300  for    2  mo.  =  $1  for    600  mo.  „o  term  of  credit  for 

400  for    4  mo.       $1  for  1600  mo.  that  sum.    Since 

81100  2200  mo.  $300  was  to  be  paid 

in  2  mo.,  the  use  of 
2200  mo. —1100=2  mo.    Average  term  of  credit,  that  sum   for  2  mo. 

i-  equal  to  the  use 
of  $1  for  600  mo.,  and  the  use  of  $400  for  4  mo.  ix  equal  to  the  u.sc 
of  $1  for  1600  months.  Hence,  the  credit  of  the  whole  debt,  $1100, 
18  equal  to  the  crcilit  of  $1  for  2200  mo.,  or  $1100  for  ^^55  part  of 
2200  mo.,  which  U  2  months,  the  average  term  of  credit. 

Rule. — Multiply  eoc/i  debt  by  its  term  of  credit,  and  divide 
the  mm  of  Hke  products  by  Uie  mm  of  tJie  debts.  The  quotient 
win.  be  the  average  term  of  credit. 

2.  H.  B.  Claflin  &  Co.  sold  a  bill  of  goods  amounting  to 
$2300,  on  the  following  terms:  $300  cash,  81200  due  in  3 
months,  and  the  balance  due  in  4  months.  What  was  the 
average  term  of  credit. 

3.  Field,  Leiter  &  Co.  sold  a  bill  of  goods  payable  as  fol- 
lows: $500  in  1  month,  $500  in  2  months,  and  $800  in  4 
months.     What  was  the  average  term  of  credit? 

4.  Whitney  &  Co.  sold  a  bill  of  lumber  on  the  following 
terms:  $1500  cash,  $3000  payable  in  30  days,  and  $2000 
l)ayable  in  90  days.  At  what  time  will  the  debt  be  payable 
in  one  cash  payment? 

5.  H.  K.  Thurber  &  Co.  sold  to  F.  N.  Burt  a  bill  of  goods 
amounting  to  S2400,  payable  as  follows:  ^  in  30  days,  ^  the 
remainder  in  60  days,  and  the  balance  in  4  months.  What 
was  the  average  term  of  credit? 

6.  Mr.  Birge  bought  a  bill  of  goods  amounting  to  83000, 
payable  as  follows:  \  m  3  mo.,  \  in  2  mo.,  and  the  rest  in 
4  mo.     What  was  the  average  term  of  credit  ? 


AVEUAOE   OF    TAVMIOMS.  293 


CASE  II. 

4(>4.  Wlieii  the  terms  of  credit  begin  at  diflereiit 
<laleK. 

1.  Find  the  average  time  of  piiynient  of  tlic  following 
bills:  Feb.  10,  1877,  $400  due  in  2  mo.;  March  15,  1877, 
8350  due  in  3  mo. ;  and  April  12,  1877,  $300  due  in  3  mo. 

PROCESS.  Analysis. 

S40U  .u.  A,,„i  10. 400  r oft,::,,;.':! 

350  due  June  15.     350x66  =  23100         chase   of   each 
300  due  July    12.  _300  X  93  =  27900         bill  its  term  of 

1050  51000         ^''^^^^'  '^'^   *'^'- 

tain    the    time 

51000  -=-  1050  =  48ff  days.  when  it  is  due, 

April  10  -f  49  days  =  May  29,  average  term,     and  so  we  have 

$400  due  April 

10,  $350  due  June  15,  $300  due  July  12.     The  average  time  when 

the  bills  will  he  due  will  be  either  after  the  earliest  date,  or  before 

the  latest  date,  and  so  we  may  select  cither  of  those  dates  from  which 

to  compute  the  average  time.    Selecting  the  earliest  date,  we  find  that 

$350  was  due  GG  days  after  that  time,  and  $300  was  due  03  days 

thereafter.     Averaging,  as  in  Case  I,  we  find  the  term  of  credit  to  be 

48\f,  or  a  fraction  more  than  48  days,  which  must  be  49  days.    This, 

ai  Ir.l  I  .  April  10,  gives  May  29,  the  average  time  of  payment. 

Rule. — SeM  the  earliest  date  at  which  any  debt  becomes  due 
for  the  standard  date,  ami  find  how  loufj  offer  that  date  tiie  otiier 
amounts  become  due. 

Find  the  average  term  of  credit  by  multiplying  each  debt  by 
the  number  of  days  from  the  standard  datr,  and  dividing  the  sum 
%f  the  prtHlucts  by  the  sum  of  tJi&  debts. 

Add  the  average  term  of  credit  to  the  >if(indard  date,  and  the 
result  will  be  tlie  average  term  of  payment. 

Inatead  of  the  earlitd  date,  they!r«<  (f  the  month  may  be  usctl. 


'-.M  AVERA«.i.    t'i     i'AYMENTS. 

2.  What  is  the  average  time  at  which  the  iuJ lowing  bills 
become  due:  Feb.  1,  1877,  $200  on  1  mo.  credit;  March  10, 
1877,  $500  on  3  mo.  credit;  April  12,  1877,  $275  on  2  m... 
credit;  and  May  1,  1877,  S400  on  4  mo.  credit? 

8.  A  merchant  owes  bills  dated  as  follows:  Jan.  1,  1877, 
$500  due  in  2  mo.;  Jan.  15,  1877,  $850  due  in  3  mo.; 
Feb.  20,  1877.  8375  due  in  3  mo.;  and,  Feb.  28,  1877, 
$650  due  in  1  n,  >.  What  will  be  the  average  time  of  pay- 
ment? 

4.  A  merchant  purchased  goods  of  Cragin  Bros.  &  < 
follows:  Sept.  10,  1876,  $300  on  4  mo.  credit;  Oct.  15,  1876, 
$400  on  6  mo.  credit;  Nov.  1,  1876,  $750  on  2  mo.  credit; 
and,  Nov.  15,  1876,  $300  on  1  mo.  credit.     AVhat  was  the 
average  time  of  payment? 

5.  Messrs.  J.  KorlMich  &  Son  Iwught  goods  from  George 
C.  Buell  &  Co.  as  follows:  Sept.  1,  1876,  $600  on  3  mo. 
credit;'  Oct.  3,  1876,  $400  on  4  mo.  credit;  Oct.  20,  1876, 
S250  on  2  mo.  credit;  and,  Nov.  10,  1876,  $375  on  1  mo. 
credit.     What  was  the  average  time  of  payment? 

6.  Stevens  &  Shepard  bought  goods  from  the  Russell  Ir- 
win Manufacturing  Co.  as  follows:  Dec.  10,  1876,  a  bill  of 
$460  on  4  mo.  credit;  Jan.  5,  1877,  a  bUl  of  $200  on  3  mo. 
credit;  Jan.  30,  1877,  a  bill  of  $200  on  4  mo.  credit;  and, 
Feb.  25,  a  bill  of  $900  on  2  mo.  credit.  What  was  the 
average  time  of  payment? 

7.  Bought  goods  of  Carson,  Pirie  &  Co.  as  follows:  Jan. 
25,  1877,  $850  on  4  mo<  credit;  Feb.  15,  1877,  $600  on  3 
mo.  credit;  March  20,  1877,  $500  on  4  mo.  credit;  and, 
April  10,  1877,  $960  on  2  mo.  credit.  What  was  the  aver- 
age time  of  payment? 

8.  May  1,  1877,  Mr.  S.  purchased  goods  to  the  amount  of 
S2400  on  the  following  terms:  J  payable  in  cash,  \  payable 
in  2  months,  and  tlie  balance  in  6  months.  When  may  the 
whole  be  equitably  paid  by  one  payment? 


AVElJAliK    UK    ACC'ULNTS. 


295 


AVERAGE  OF  ACCOUNTS. 

405.     1 .  What  should  be  the  date  of  a  note  given  to  settle 
tlie  tbllowiug  account? 


Dr. 


W.  11.  8ij:vENS. 


Pkocfxs.     {By  Products.) 


Or. 


1877. 
May 

5 

To  Mdsc. 

50 

00 

1877. 
May 

15 

By  Cash 

25 

00 

June 

7 

"     2  mo. 

140 

00 

June 

10 

"  Draft,  10  da. 

100 

00 

June 

20 

..        ..       I    u 

'^ 

00 

June 

30 

4(              it 

100 

00 

Dne. 


May 
Aug, 
July 


S50 
140 
150 


225 


115 


Dnif$. 


Proilnct. 


4700 


2550 


7250 


Puul. 


May 
June 
June 


15 


Anouut. 


$25 
100 
100 


/>!*«!  I  i 


3150-^  115  =  27^<V 


2100 
4  "00 
3.SJ0 

lOlUO 
Ti->0 


3150 


Aug.  7  +  28  days  =  Sept.  4,  the  average  time. 

Analysis.— From  the  dates  at  which  the  variou.s  amouiiLs  become 
due,  wc  select  the  latest,  which  is  Aug.  7,  for  the  a.ssumed  time  of  set- 
tlement, and  multiply  each  amount  by  the  number  of  days  intervening 
between  that  date  and  the  time  when  each  item  of  tiie  account  becomes 
due.  The  debit  side  of  the  accoimt  shows  there  is  due  $340  and  the 
use  of  $1  for  7250  days,  and  the  cretlit  side  shows  that  $225  has  been 
paid,  and  that  the  debtor  is  entitled  to  the  use  of  $1  for  10400  days,  if 
the  time  of  settlement  is  Aug.  7.  Subtracting  the  amounts,  there  is 
shown  to  l)e  $115  due,  and  the  debtor  is  entitled  to  the  use  of  $1  for 
3150  days.  Tliercfore,  he  should  not  be  re(jnire<l  to  pay  the  account 
until  the  time  when  the  use  of  $115  U  equal  to  the  ust»  of  $1  for  3150 
dayK,  which  is  28  days.     28  days  after  .\ug.  7  i*  Sept.  4. 


290 


AVJ:i;AGK  « 'I 


Rule. — Multiply  each  anumni  um  uj  atr  itmnhrr  nj  days  in- 
tervemng  between  the  time  it  becovies  due  and  the  latest  date  at 
which  any  ^im  on  either  side  of  the  account  becomes  due. 

Divide  the  difference  between  the  mm  of  the  products  of  Vie 
debit  and  credit  side  of  tlie  accounty  by  the  balance  due  on  the 
account.     The  quotient  wUl  be  tfie  average  term  of  credit. 

1.  When  the  balances  arc  bolli  on  one  side  of  the  account,  the  term 
of  credit  is  to  be  counted  bachuird  from  the  date  at  which  the  first 
amount  bccomefl  due,  but  forward  from  that  date  if  the  balances  are 
on  ()pi)osite  Rides. 

2,  The  average  term  of  credit  may  also  be  found  by  reckoning 
interest  upon  each  sum  due  for  tlie  number  of  days  intervening  be- 
tween the  time  it  becomes  due,  and  the  earliest  date  at  which  any 
pum  becomes  due;  then  dividing  the  balance  of  the  interest  by  the 
interest  on  the  balance  of  the  account  for  one  day.  This  is  called  the 
Mdhod  by  Interest.  The  result  is  the  same  whether  the  average  term 
of  credit  is  found  by  the  method  by  products  or  by  interest. 

2.  Find  the  average  term  of  credit  of  the  following  account : 
Dr.  Olmstead  &  Bishop.  O. 


1S77. 

1877. 

Jan. 

5 

To  Mdse.,2mo. 

$375 

Jan. 

30 

By  Cash 

$200 

Feb. 

15 

..   ..   1  .. 

200 

Mar. 

15 

"   •• 

600 

Feb. 

25 

4  " 

800 

Apr. 

1 

<>   II 

200 

Mar. 

:» 

"   "   3  •• 

450 

3.  Find  by  both  methods  when  the  balance  of  the  follow- 
ing account  becomes  due. 

Dr.  Hamilton  &  Matthews.  Or. 


1876. 

1  1877. 

Nov. 

1 

To  Mdse.,  4  mo. 

$1600 

Jan. 

15 

By  Accept'ee,  2  mo. 

S2000 

Dec. 

3 

3  " 

3800 

Mar. 

20 

2  " 

5000 

1877. 

1 

, 

Jan. 

15 

"   4  " 

5500 

Mar. 

1 

11   ..   3  II 

1503 

! 

AVERAGE   OF   ACCOUNTS.  207 

(.    ^^'hen  should  interest  begin  on  the  following  account? 
Dr.    James  Howard,  t»  ojcc't  wWi  Hiram  Sibley.    Or, 


isil. 

1877. 

Apr. 

10 

To  Mdso. 

;loO 

Apr. 

12 

«y  Cash 

S2.y) 

Apr. 

r.o 

.. 

400 

May 

1 

•' 

•JOO 

May 

K. 

" 

100 

June 

7 

" 

4tH) 

Jmif 

•Jl 

"        " 

000 

5.  Find  the  average  term  of  credit  of  the  following  account? 
I>r.  Breyfogle  &  Co.  Cr. 


1877. 

1877. 

^ 

Jan. 

1 

To  Mdsc.,  1  mo. 

$.-00 

Feb. 

.3 

By  Cnsh 

$r,00 

Jan. 

20 

..       3  .. 

8."-0 

Feb. 

28 

•• 

200 

Feb. 

15 

..      2  •• 

IJOO 

Mny 

15 

•*  Draft,  1  nio. 

1200 

Apr. 

3 

..       ..      4  .. 

* 

2.')00 

•  '.    When  should  interest  begin  on  the  following  account? 
/>/.  BuRK,  FiTZsnioxs,  Hone  &  Co.  Cr. 


1^::. 

1877. 

Feb. 

1 

ToM  : 

SISOO 

Feb. 

20 

By  Cash 

$3000 

Mar. 

15 

„ 

3000 

May. 

18 

"  Accept' ee,  2  mo. 

8000 

Mur. 

20 

, 

4800 

Apr. 

3 

1    ■■ 

GOOD 

7.  What  will  l)e  the  cash  balance  of  the  following  account, 
Jan.  1,  1878,  interest  at  6^? 

/>/.  Pratt  J.  Nedbon.  Cr. 


is::. 

1 

J877. 

July 

10 

To  Mdse.,  2  mo. 

1500 

July 

•20 

By  Cash 

S400 

Aug. 

1 

..       ..       3    .. 

700 

Aug. 

20 

««       it 

1000 

Sept. 
Sepu 

A 

"       •'       1    •• 
M       ..       2    •• 

800 
600 

U^^T 

II  q  T 


PARTNERSHIP 


Ni^HBi^^mHBHHBiVC^ 


46().  1.  If  two  men,  who  have  equal  sums  invested  in  the 
same  business,  gain  $100,  what  is  vi\('h  nKnTs  share  of  the 
gain? 

'2.  If  one  man  furnishes  |  of  the  cajiiUil,  and  another  ^  of 
it,  and  the  gain  is  SI 200,  what  should  be  the  gain  of  each? 

3.  Mr.  A.  furnishes  $3000  of  the  capital,  and  Mr.  B.  fur- 
nishes the  balance,  which  is  $5000.  What  part  of  the  profits 
should  each  receive? 

4.  Four  partners  furnish  money  in  the  proportion  of  $2000, 
$3000,  84000,  and  $5000  respectively.  What  part  of  the 
gain  should  ciich  one  receive? 

5.  Three  men  engage  in  business  and  furnish  the  following 
sums  respectively:  A,  S5000;  B,  84000;  C,  83000.  How 
much  of  the  gain  should  each  receive  if  81200  was  gained 
during  the  year? 

6.  The  cost  of  a  pasture  was  $27.  A  had  in  it  5  cows  for 
3  weeks,  and  B  3  cows  for  4  weeks.  What  should  each  one 
pay? 

7.  The  profits  of  a  company  were  $800  for  a  certain  time. 
AVhat  share  of  the  profits  did  each  partner  receive,  if  the 
capital  contributed  by  them  was  $900,  $700,  and  $800  re- 
s])ectively? 

8.  A  and  B  formed  a  partnership  after  A  had  been  doing 
business  alone  for  6  months.  A  had  $5000  invested  during 
the  year,  and  B  had  $10000  invested  for  6  months.  The 
gain  was  $5000.     What  was  each  one's  share? 

(298) 


rAKTM':KJ5Mir.  .  299 


DEFINITIONS. 

4G7.  A  Partnership  is  an  association  of  two  or  more 
persons,  for  the  purpose  of  conducting  business. 

468.  I*{irt tiers  are  the  persons  associated  in  business. 
Tliey  are  called  collectively  a  company,  a  Jinn  or  house. 

469.  The  Capital  is  the  money  employed  in  business. 

470.  Principle. — Partners  share  the  gains  and  losses  in  pro- 
poHion  to  the  amount  of  the  capital  each  invests,  and  tJie  lengtli 
of  time  it  is  employed. 

CASE  I. 

471.  When  the  capital  of  eaeh  partner  in  eiiiploye«1 
lor  th<*  same  lime. 

n  /.•  /  I  r  i:  v    /; XEltCISES. 

1.  A,  B  and  C  are  partners,  having  furnished  65000, 
$6000  and  88000  capital  respectively.  If  during  the  year 
they  gain  §2850,  what  is  each  partner's  share  of  the  gjiin? 

PROCESS.  Analysis. 

$5000  -f  S6000  -f-  S8000  =  S19000  ^'"f  ''X\Tui 

t^Mf  or  tV  ^^  $2850=    S750,  A's  share.         share  the  gain 

VWAftr  or  A  of  $2850=  $900,  B's  share.        in  proportion  to 
iwuwu  lu  ll^g  amount  of 

VyW/V  or  ^  of  $2850=81200,  Cb  share.  eapital  he  con- 
$2850,  Whole  gain.  tributcd,wcfina 
what  part  of  the 
whole  capital  each  partner  contribute*!.  A  furnished  i\  of  the  eapital, 
and  ist  theref«»re  entitled  to  /,j  of  the  gain,  or  $750.  B  furnished  ^^ 
of  the  eapital,  and  is  entitle<l  to  y".j  of  the  gain,  or  $*.)0(),  C  furnishe*! 
>^,  of  the  capital,  ami  is  therefore  entitled  to  ^'j  of  the  gain,  or  $1200. 


300  PARTNERSHIP. 

UcLE. — Find  such  a  part  of  Vie  (jam  or  Utm  cts  ccwli  partnered 
citpiUd  w  of  the  whole  capital. 

The  reauU  will  be  each  partner's  gain  or  lose. 

2.  A,  B  and  C  engaged  in  business,  employing  $20000 
capitiil,  of  which  A  furnished  87000,  B  S7000,  and  C  $0000. 
They  gained  in  one  year  $6000.  What  was  each  partncr'.s 
sliare? 

3.  Three  men  engage  in  Imsincss.  A  furnishes  $3000  of 
the  capital,  B  $0000  and  C  $4000.  If  tliey  gain  $2000,  what 
is  each  partner's  share? 

4.  Three  men  engaged  in  hind  siK'cuhition.  A  i'urnislicd 
$10000,  B  $8000  and  C  $12000.  They  lost  in  one  year 
$6000.     What  was  the  loss  of  each  partner? 

5.  A,  B  and  C  furnish  capital  to  engage  in  l)U.siuess  as 
follows:  A  $2500,  B  $2000  and  C  $3500.  If  the  firm  loses 
$640, what  is  the  loss  of  each  partner? 

6.  A,  B,  C  and  D  engaged  in  buying  produce.  A  con- 
tributed $8000,  B  $10000,  C  $9000  'and  D  $13000.  They 
gained  $3000.     What  was  each  jmrtner's  share  of  the  gain? 

7.  D  and  G  furnish  ca})ital  to  engage  in  business  and  L 
does  the  work  for  ^  of  the  profits;  D  contributes  $8000  and 
G  10000  of  the  capital.  They  gain  $5400.  AVhat  is  each 
one's  share  of  the  gain  ? 

8.  E,  F  and  G  bought  a  block  of  stores  for  $46000.  E 
furnished  f  of  the  money,  F  $11500  and  G  the  rest.  The 
l)roperty  was  sold  for  $48300.     Wliat  was  the  gain  of  each  ? 

9.  A,  B  and  C  engage  in  business.  A  furnishes  $6470, 
B  $5420  and  C  $3410  capiUil.  If  they  gain  $6490.75,  what 
is  the  gain  of  each  ? 

10.  Four  persons  rented  conjointly  a  pasture  containing 
125  A.  60  sq.  rd.,  for  83.75  an  acre.  A  fed  125  sheep  upon 
it,  B  145  sheep,  C  175  sheep,  and  D  340  sheep.  How  much 
rent  should  each  one  lay? 


r A  lii.NEU.su  11'.  301 

11.  Three  iikmi  (Mij^agt'd  in  business.  A  fiiriii^'mtl  ^GOUO 
and  B  88000.  They  gained  $4200,  of  which  C's  share  was 
$1400.     Wiiat  was  the  gain  of  A  and  B  and  C's  stock? 

12.  Five  men  trade  in  partnership.  A  furnishes  8500, 
B  S600,  C  $800,  D  $1000  and  E  $1200  capital.  They  gain 
$2750.     Wliat  is  the  gain  of  each  partner  ? 

13.  A,  B  and  C  bought  a  farm  in  partnership.  A  paid  J 
the  purchase  money,  B  J  and  C  the  rest.  They  sold  it  at  a 
gain  of  $3000.     What  was  each  one's  share  of  the  profit? 


CASE  II. 

472.  When  the  capital  of  tlio  partners  is  employed 
Ibr  Uiirereut  periods  ol'tiiiie. 

WBITTEX    EX  ETtCIS  ES. 

1.  A  began  business  with  $6000  capital.  At  the  end  of 
()  montlis  he  took  in  B  as  a  partner,  who  furnished  $5000 
achlitional  capital.  If  the  gain,  after  G  months  more,  was 
$3400,  what  was  each  partner's  share  of  the  gain? 

PROCESS.  Analysis. — A'scap- 

^««^^       -^     \^r.^^r.  ilal  of  $6000  was  ust'd 

$6000X12  =  $72000  forl2.nonlhs,and  was 

$5000  X     6-$30000  Ihcrefore  equal  to  the 

$102000  "se  of  $72000  for    1 

month.      B*s    capital 

T^^  of  $3400  =  $2400,  A's  share.      of  $5000  was  used  for 

^%  of  $3400=--$  1000,  B's  share.       C  months,  which  was 

equal  to  the  use  of 
$30000  for  1  month,  lloth  to^athLr  had  invested  sums  of  money 
whieh  were  equal  to  the  une  of  $102000  for  1  month,  of  which  A  con 
tribnted  a  »um  equal  to  $72000  for  1  month,  or  ^Vi»  *"d  he  was  there- 
fore cntitliHl  to  tV,  of  the  gain,  or  $2400.  B  contributed  a  sum 
equal  to  $30000  for  1  month,     ;     '"        -  1  V  ;    *  tithd  to 

1*0^2  of  the  gain,  or  $1000. 


302  PAUTNERSHIP. 

Rule. — Find  such  a  part  of  the  entire  gain  or  loss,  for  each 
partner's  share  of  Hie  gain  or  loss^  as  the  capital  of  each  partner 
for  a  unit  oftiiney  is  ofOie  entire  capital  for  a  unit  of  time. 

2.  A  engag^  in  business  with  a  capital  of  §4000.  After 
3  months  he  took  in  B  with  a  cupitiil  of  SGOOO,  and  in  6 
more,  C  became  a  jwirtner,  with  a  capital  of  $8000.  At  the 
end  of  18  months  the  profits  were  $9360.  What  was  each 
partner's  share  of  the  gjiin  ? 

3.  A,  B  and  C  engage  in  business  together.  A  puts  in 
84000  capital  for  8  months,  B  $6000  for  7  months,  and  D 
Sa.lOO  for  one  year.  If  thrv  -niii  ^2'V1^\  wluit  is  each  part- 
ner's share  of  the  gain? 

4.  B,  C  and  D  entered  into  partnership,  furnishing  a 
joint  capital  of  $5875,  of  which  B  furnished  20%,  C  35%, 
and  D  the  rest.  B's  capital  was  employed  15  months,  C's 
9  months,  and  D's  10  months.  They  lost  $2502.75.  What 
was  each  partner's  loss? 

5.  A,  B  and  C  took  a  contract  to  build  a  block  of  stores. 
A  furnished  20  men  for  3  months,  B  25  men  for  3J  months, 
and  C  15  men  for  4  months.  Aft«r  paying  the  expenses 
the  profits  were  $1475.     What  was  the  share  of  each? 

6.  A,  B  and  C   lost  $8500   by  speculating  in  real  estate. 

A  furnished  $5000  of  the  capital  which  was  employed  for  1      ^ 
year,  B  $8000  for  10  months,  and  C  810000  for  6  months. 
What  was  each  one's  share  of  the  loss? 

7.  A,  B  and  C  engaged  in  manufacturing  rope  and  cord- 
age.    A  invested  §4500  for  6  months,  B  $5000  for  8  months,  ' 
and  C  86500  for  7  months.     They  gained  84500.     What  was 
the  gain  of  each  partner? 

8.  G,  L  and  F  entered   into   partnership.     G  furnished 
81200,  L  81500,  and  F  83000.     After  6  months  F  withdrew-     ^ 
82000  of  his  capital.     If  at  the  end  of  a  year  the  profits  were   - 
$2200,  what  part  of  the  profits  belonged  to  each  partner? 


\X7 


473.  1.  A  was  employed  on  a  piece  of  work  6  days,  and 
B  12  days  on  the  same  work.  How  docs  the  number  of 
(hiys  A  worked  compare  with  the  number  of  days  B  was 
employed? 

2.  A  laborer  earned  §12  a  week,  and  spent  86.  How 
decs  what  he  si)ent  compare  with  what  he  earned  ^ 

:i  How  <16es  $3  compare  with  $9?  $4  with  $12?  66 
with  $18? 

4.  How  does  2  compare  with  10?  3  with  18?  5  with 
2:.  ? 

5.  What  relation  has  2  to  12?    3  to  21?     4  to  28? 

6.  What  is  the  relation  of  3  to  24?     6  to  30?     7  to  35? 

7.  How  does  8  compare  with  2?  What  is  the  rolatioTi 
of  8  to  2? 

8.  How  does  9  compare  with  3?  What  relation  has  9 
to  3? 

9.  What  relation  has  24  to  8?    30  to  6?    25  to  4? 

1 0.  What  is  the  relation  between  5  and  7  ?     Ans.  ^  or  J. 

11.  What  is  the  relation  between  6  and  9?  Between  8 
and  9? 

12.  What  in  the  nlaiion  of    8  to  9?     Between    8  and  9? 

13.  What  is  the  relation  of  12  to  4?     Between  12  and  4? 

14.  What  is  the  relation  of  15  to  5?     Between  15  and  5? 

15.  What  is  the  relation  of  16  to  8?     Between  16  and  8? 

16.  What  is  the  relation  of  25  to  5?    Between  25  and  5? 

(308) 


o04  RATIO. 


DEFINITIONS. 

474.  Mat  to  is  the  relation  of  one  number  to  another  of 
the  same  kind. 

1.  This  relation  ia  cxpressetl  cither  as  quotient  of  one  number 
divided  by  tlje  other,  and  is  called  Geometrical  Ratio,  or  simply  Ratio, 
or,  as  the  difference  between  two  numbers,  and  is  called  Arithmetical 
JliUio. 

2.  When  it  is  required  to  determine  wIuU  the  relation  of  one  number  to 
v.nothtr  {".  '"  '-  '■^i'N'iit  th:\t  tho  fhy(  i«  t!"-  •livl'?"'vl,  nnd  tho  fsem-.-l  »!.». 
divisor. 

3.  WIku  It  is  1.  .  ,:i.  I  to  delormiiu' f//'-  vini^  (iro  nmnUrrs, 
tiihcr  may  be  rega;  ;    ;     -  dividend  or  divisor. 

4.  The  first  number  is  commonly  regarded  as  thu  dividend. 

475.  The  Terms  of  a  Hatio  are  the  nunibci-s  com- 
l).\rcd. 

476.  The  Antecedent  is  the  first  term. 

Thus,  in  "What  is  the  ratio  of  6  to  8?"  6  is  the  antecedent. 

477.  The  Consequent  is  the  second  term. 

Thus,  in  "  What  is  the  r^io  of  6  to  8  ?"  8  is  the  consequent. 

478.  The  Sign  of  ratio  is  a  colon  (:). 
Thus,  the  ratio  of  12  to  6  is  expressed,  12  :  6. 

The  colon  ( : )  is  sometimes  regarded  as  the  sign  of  division  without 
the  line.    Thus,  12  :  8  is  regarded  as  12 -r- 8. 

479.  The  antecedent  and  consequent  together  form  a 
Coniflet. 

480.  Principles. — 1.  The  terms  of  a  ratio  mud  be  like 
numbers. 

2.  The  ratio  is  an  abstract  number. 

3.  Multiplying  or  dividing  both  terms  of  a  ratio  by  Hie  same 
number  does  not  change  Hie  ratio  of  Hie  numbers. 


RATIO.  30.J 


EXERCISES. 


4<S1.    1.  What  is  the  ratio  of  o  tu  «j  ?     5  to  10?     7  to  21  ? 

2.  What  is  the  ratio  of  $3  to  $10?     12  lb.  to  6Jb.?     27 
Ini.sh.  to  9  bush.? 

3.  What  is  the  ratio  of  7  to  35?     li 4  lo  48?     13  to  39? 

4.  If  the  antecedent  be  20,  and  the  consequent  15,  what  is 
the  nitio? 

').  What  is  the  ratio  when  the  antecedent  is  45,  and  the 
oonsequent  25? 

(>.  What  is  the  ratio  of  |  to  |?     §  to  }?     %  to  ^? 

Fractions  should  be  reduced  to  similar  fracti<^ns,     TJiov  will  then 
have  the  ratio  of  their  numerators. 

7.  What  is  the  ratio  of  5}  to  SJ?    7|  to  6}?    9 J  to  5 J? 

8.  What  ratio  will  the  work  of  12  men  sustain  to  that  of 
^  men? 

9.  What  will  Ikj  the  ratio  of  8  yd.  to  24  yd.?     6 J  yd.  t  . 
Jyd.? 

10.  When  the  antecedent  is  3  and  the  ratio  J,  what  Is  the 
'  onsequcnt? 

11.  When  the  consequent  is  8  and  the  ratio  5.  what  is  tlie 
antecedent  ? 

12.  When  tijc  antecedent  is  }  and  the  ratio  \,  wlrat  is  the 
consequent? 

1:^  What  numl)er  has  to  3  the  ratio  of  5  to  6? 

14.  What  numlxir  has  to  5  the  ratio  of  4  to  12? 

15.  What  number  has  the  ratio  to  12  that  3  has  to  1  ' 
IT).  If  two  numl)ers  have  the  relation  of  6  to  8,  and  tlie 

lii-st  is  12,  what  is  the  other? 

17.  Wliat  number  luis  to  12  the  ratio  of  8  to  9? 

18.  If  two  numlxTs  have  tlie  relation  of  10  to  15.  and  the 
ntecedcnt  is  40,  what  is  the  consequent? 

20 


Att 


iHiiimmT 


^JrAk 


PROPORTION    t 


482.  1.  What  two  numbers  have  the  same  relation  to  each 
other  as  3  to  6?     As  2  to  8?     As  7  to  21? 

2.  What  two  numbers  have  the  same  ratio  ns  5  to  15? 
6  to  30?     12  to  48?     12Jto25?    2J  to  4^?     12^  to  50? 

3.  What  number  has  the  same  relation  to  6  that  3  has 
to9? 

4.  What  number  has  the  same  relation  to  5  that  7  has 
to  14? 

5.  What  number  has  the  same  relation  to  f  that  4  has 
to8? 

G.  To  what  number  has  5  the  same  relation  that  3  has 
to9? 

7.  To  what  number  has  2^  the  same  relation  that  7  lias 
to  21? 

8.  24  is  to  7  as  12  is  to  what  number? 

9.  12  is  to  5  as  what  number  is  to  15? 

10.  If  the  cost  of  9  yards  of  cloth  is  S5,  how  will  the  cost 
of  18  yards  compare  with  that  sum? 

11.  If  10  men  can  earn  S30  per  day,  what  ratio  will  the 
earnings  of  15  men  bear  to  that  sum? 

12.  Write  two  equal  ratios;  multiply  the  first  and  last 
terms  together ;  multiply  the  second  and  third  terms  together. 
How  do  the  products  compare? 

13.  Write  two  other  equal  ratios;  multiply  as  before. 
How  do  the  products  compare? 

(306) 


PROl'ORTION.  307 


DEFINITIONS. 

4<SJ}.  A  Proportion  is  an  equality  of  ratios. 
Thu3,  9  :  18  =  6  :  12  is  a  proportion. 

484.  The  Sign  of  proportion  is  a  double  colon  (  :  :  ). 
The  double  colon  ( :  : )  may  be  regarded  as  the  extremities  of  the 

sign  of  equality  (  =  ).    It  is  written  between  the  ratios. 

A  proportion  must  have  four  terms,  viz:  two  antecedents, 
and  two  consequents. 

Any  four  numljers  that  can  be  formed  into  a  proportion 
are  called  projjortioiials. 

485.  The  Antecedents  of  a  proportion  are  the  antece- 
dents of  the  ratios,  or  the  first  and  third  terms. 

Thus,  in  the  proportion  5  :  10  :  :  7  :  14,  5  and  7  are  the  antecedents. 

486.  The  Consequents  of  a  proportion  are  the  conse- 
quents of  the  ratios,  or  the  second  and  fourth  terms. 

Thus,  in  the  proportion  5  :  10  :  :  7  :  14,  the  consequents  are  10  and  14. 

487.  The  JExtreines  of  a  proportion  are  the  first  and 
fourth  terms. 

Thus,  in  the  proportion  7  :  8  :  :  14  :  IG,  7  and  16  are  the  fxlremes. 

488.  The  Means  of  a  proportion  are  the  second  and 
third  terms. 

Thus,  in  the  proportion  7  :  8  :  :  14  :  IG,  8  and  14  are  the  mean*. 

489.  Principles. — 1.  The  product  of  the  ejrtremeji  u  equal  to 
the  product  of  the  means. 

2.  The  product  of  the  extremes  diviil"!  !>>/  (iihrr  mnm  gives 
ihe  oOier  mean. 

3.  The  product  of  tJie  meam  divided  hij  eiOier  extreme  gives  ihe 
other  extreme. 


308 


i'i:< )!'( )irri«).\. 


EXSBCI8E8, 


Find  the  term  that  is  wantinj^  in  the  following: 


1. 

18 

:24  : 

;     I  , 

:•. 

:    1  I  ::    li;  :  :;:.. 

2. 

9: 

18:: 

( ; :  ' 

10.   14:  23::  (  ):  69. 

3. 

8  : 

18  :: 

7  :  ( 

11.   13:  (  )  ::ir>  :  fir> 

}, 

:    1^ 

:  :    7   :    I  •">. 

12.  i 

:5::(): 

5. 

7  ; 

n-' 

:  8  :  24. 

13.1 

()::i:10. 

6. 

15 

:  18  : 

:  (  )  :  16. 

14.  i 

|::():15. 

7. 

17 

:   1!»  : 

:  15  :  (  ). 

15.  6 

i::5:(). 

8. 

2:> 

in  :  25. 

16.   13:  7  ::  (  )  :  8. 

1  i . 

.)  men  :  7  men  : :  8J  :  (  ). 

18. 

821.16  :  $1 

3.20::4:(). 

19. 

80.51  A.  :  21.15  A: :  (  )  :  2. 

20. 

16  lb.  3  oz. 

:  18  lb.  2  oz.  : :  (  )  :  7. 

21. 

5  gal.  3  qt. 

:(  )::5:9. 

22. 

14%  :  (  )  : 

:  6  :  15. 

SIMPLE  PROPORTION. 

400.  A  Simple  Hatio  is  a  ratio  between  any  two 
numbers. 

Thus,  6:8,  $10: $8,  5  lb.  6  oz.:  7  lb.  3  oz.,  are  fiiraple  ratios. 

491.  A  Simple  JProjyortion  is  an  equality  between 
two  simple  ratios. 

492.  A  Direct  I^roporfion  is  one  in  which  each 
term  increases  or  diminishes,  as  the  one  on  which  it  depends 
increases  or  diminishes. 

Thus,  proportions  involving  quanl.Uy  and  cost,  men  and  work  done, 
etc.,  are  direct  proportions,  for  as  the  quantity  increases  or  diminishes, 
Ihe  cost  increases  or  diminishes,  and  as  the  number  of  men  increases  or 
diminishes,  the  amount  of  work  done  will  ina-ease  or  diminish. 


PROPORTION.  309 

493.  An  Jnrerse  Proi)Ortioit  is  one  in  wliich  each 
term  incrt:ai<cs  as  the  term  U{X)n  which  it  dei)euds  diminisliesy 
or  dimini^s  as  it  increases. 

Thus,  in  the  problem,  "  If  6  men  can  mow  a  field  of  grass  in  9 
days,  how  long  will  it  take  9  men  to  mow  it,"  as  the  number  of  men 
increases,  the  number  of  days  required  to  do  the  work  dca'eaaeSf  and  the 
proportion  is  an  inverse  proportion. 


WRITTEN   EXERCISES. 

494.    1.  If  8  yd.  of  silk  cost  324,  what  will  15  yd.  coat? 
PROCESS.  Analysis.— It  is  evi- 

yd.    vd.       $        $  ^^"'  ^''^*  ^  y^-  ''^^'^  *^® 

(1)  8  :  15  : :  24  :  (  )  sa^e  relation  to  15  yd. 

that  the  cost  of  8  yd.  has 

/ON      1^"    ^f'      A      ^4  to  the  cost  of  15  yd.  Hence 

(2)  15  :  8  ::  (  )  :  24  .         ,, 

^  ^  ^  ^  we  have  the  proportion, 

The  term  wanting  (1)  -L5^  ^^  345       g  yd.  :  15  yd.  ::  $24,  the 
The  term  wanting  (2)  -'^^4^  =  $45       cost  of  8  yd. :  the  cost  of 

15yd.,  or  15 yd.  :8  yd.  :: 
the  cost  of  15  yd.  :  $24,  the  cost  of  8  yd.  To  find  the  cost  of  15  yanls, 
the  term  wanting,  we  divide  the  product  of  the  means  by  the  extreme, 
aK  in  (1) ;  or  the  product  of  the  extremes  by  the  mean,  as  in  (2). 

2.  If  5  men  can  cut  a  quantity  of  wood  in  18  days,  in  liow 
many  days  could  12  men  do  tlie  same  work? 

PROCESS.  Analysis. —  It   is   evident 

men.  men.     days.  days.  ^^»«^  ^^"^^^1^  »"  proportion  as 

(1)  5  :  12  ::  (  )  :  18  *^®  number  of  men  is  inermaedj 

the  nunilxT  of  days  required  to 

(2)  T2"'-"5"'-  is"-  m"**  ^"^  ^'**'  ''''"*''  '^  ^»"'""'''*^^'  «"'> 
^   ■'          '       ' '         '  \  ''  therefore  5  men  :  12  men  ::  the 

Term  wanting  =  ^-fyi  =  7<J  da.      days  it  will  require  12  men  to 

do  the  work  :  18  the  number  of 

(iavB  required  for  5  men  to  do  the  work.    Or, 

12  men  :  6  men  ::  18  days,  the  number  of  days  it  requires  5  men  lo 

do  the  work  :  the  number  of  days  12  men  require  to  do  the  work. 


310  ^  PROPORTION. 

Rule. — Express  Vie  ratio  between  the  two  numbers  that  are  like 
numbers.  Consider,  from  the  conditions  of  the  problem^  whether 
the  proportion  is  direct  or  indirect,  and  arrange  the  other  number 
and  the  term  tuanted  so  thai  the  tux)  ratios  will  be  equal. 

Divide  the  product  of  Vie  extremes  or  means  by  Vie  single  ex- 
treme or  mean,     Tlie  result  will  be  the  term  wanted. 

Problems  in  proportion  are  sometimes  regardcnl  as  illustrations  of 
cause  and  effect,  in  which  two  causes  and  their  corresponding  effects 
are  compared,  giving  the  following  proportion: 

1st  cause  :  2d  cause  : :  Ist  effect :  2d  effect. 

3.  If  6  men  earn  $75  in  one  week,  bow  much  will  10  men 
earn  in  the  same  time? 

4.  If  16  yards  of  cloth  cost  $20,  what  will  be  the  cost  of  7 
yards? 

5.  A  man  am  buy  45  sheep  for  3112.50.  How  much  will 
18  sheep  cost  at  the  same  rate? 

6.  If  8  horses  consume  15  tons  of  hay  in  6  months,  how 
much  hay  will  14  horses  consume  in  the  same  time? 

7.  If  6  men  can  do  a  piece  of  work  in  45  days,  how  many 
days  will  it  take  11  men  to  do  the  same  work? 

8.  If  10  men  can  do  a  piece  of  work  in  6  days,  in  how 
many  days  can  13  men  do  the  same  work? 

9.  How  many  men  will  it  require  to  build  60  rods  of  wall 
in  the  same  time  that  8  men  can  build  40  rods? 

10.  If  8  men  can  dig  a  ditch  in  15  days,  how  many  days 
will  it  take  13  men  to  dig  it? 

11.  If  6  bushels  of  wheat  can  be  bought  for  $7.32,  how 
many  bushels  can  be  bought  for  $45? 

12.  How  many  barrels  of  apples  can  be  bought  for  $2250, 
if  15  barrels  cost  $33.75? 

13.  K  it  requires  13  men  to  lay  a  certain  number  of  bricks 
in  28  days,  how  many  days  will  it  take  9  men  to  lay  the  same 
number  ? 


COMPOUND    PROPORTION.  311 

14.  If  165  bushels  of  potatoes  can  be  raised  on  1^  acres 
of  ground,  how  many  bushels  can  be  raised  on  3^  acres? 

15.  If  it  requires  H  acres  of  ground  tx)  raise  405  bu.  of 
carrots,  how  many  acres  will  it  require  to  raise  975  bu.  ? 

16.  Five  horses  cost  a  man  §626.25.  What  would  be 
the  cost  of  13  horses  at  the  same  rate? 

17.  It  required  26  men  to  build  an  embankment  in  80 
days.  How  long  would  it  require  32  men  to  do  the  same 
work? 

18.  It  took  9  horses  to  move  a  stick  of  timber  weighing 
12590  pounds.  How  many  pounds  would  a  stick  weigh 
which  could  be  moved  by  7  horses? 

19.  If  an  qcean  steamer  sails  1775  miles  in  5  days,  how 
many  miles  will  she  sail  in  6J  days? 

20.  If  a  locomotive  runs  96f  miles  in  3^  hours,  how  many 
miles  will  it  run  in  5J  hours? 

21.  A  dog  is  chasing  a  rabbit,  which  has  45  rods  the  start 
of  the  dog.  The  dog  runs  19  rods  while  the  rabbit  runs  17. 
How  far  must  the  dog  run  before  he  catches  the  rabbit? 

22.  A  cistern  has  3  pipes.  The  first  will  fill  it  in  12 
hours,  the  second  in  16  hours,  and  the  third  in  18  hours. 
If  all  run  together  how  long  will  it  take  them  to  fill  it? 

23.  If  it  requires  15  compositors  15  days  to  set  up  a  book 
f  675  pages,  how  many  days  will  they  need  to  set  up  a  book 

<»f  900  pages? 

COMPOUND  PROPORTION. 

495.  A  Compound  Ratio  is  the  product  of  two  or 

more  simple  ratios. 

4%.  A  Compound  Propartion  is  a  proportion  in 
which  either  ratio  is  compound. 

497.  Principle.— TAe  product  of  Uoo  or  more  proportions  is 
fi  proportion. 


312 


1»R0PURT10N. 


WBITTEir   BXBBCISJJS. 


4^98.  1.  If  6  men  can  mow  24  acre.s  w.  ;.,a.'.-  in  2  liuviS, 
by  working  10  hours  per  day,  how  many  days  will  it  take  7 
men  to  mow  56  acres,  by  working  12  hours  per  day  ? 

PROCESS.  Analysis. — A  simple  pro- 


(1) 

(2) 
(3) 


(4) 


7:    6 
24:56 

12:in 


:2 

:4 
Or. 


(4    days.) 
m  days.) 


^  24 :  56  y 
U2:10J 


■.2■.^ 


Means     6  X  56  X  10  X  2 


H 


lx)rtion  is  a  proportion  that 
has  but  one  condition.  A  com- 
pound proportion  has  more 
than  one  condition.  The  con- 
ditions arc  introduced  one  at 
a  time,  therefore  examples  in 
compound  proportion  may  be 
solved  as  several  simple  pro- 
portions. The  first  condition 
in  this  example  is :  If  6  men 
can  mow  24  acres  of  grass  in 


days,  how  long  will  it  take 


Extremes     7  v  24  X  12 

7  men  to  do  the  same  work? 
This,  solved  by  simple  proportion,  (1),  gives  1^  days.  The  second 
condition  is:  If  the  men  can  mow  24  acres  of  grass  in  If  days,  how 
long  will  it  take  them  to  mow  56  acres?  This,  solved  by  simple  pro- 
Ix)rtion,  (2),  gives  4  days.  The  third  condition  is:  If  the  work  can 
be  done  by  the  men  in  4  days,  by  working  10  hours  per  day,  how 
many  days  will  it  take  to  do  the  work  if  they  work  12  hours  per  day? 
This,  solved  by  simple  proportion,  (3),  gives  3^  days,  the  time  it  will 
take  7  men  to  mow  56  acres  of  grass,  by  working  12  hours  per  day,  if 
six  men  can  mow  24  acres  in  2  days  by  working  10  hours  per  day.     Or, 

Since  every  simple  proportion  is  an  equality  of  ratios,  the  product 
of  the  three  proportions,  (1),  (2),  (3),  will  be  an  equality  of  ratios; 
and,  since  1|  and  4  appear  in  both  antecedent  and  consequent,  they 
may  be  omitted,  and  the  simple  proportions  will  assume  the  form  of 
the  compound  proportion,  (4). 

The  problem  may  be  stated,  as  in  the  second  part  of  the  process,  by 
writing  for  the  third  term  the  term  that  is  like  the  one  sought,  and  by 
arranging  the  others  in  couplets,  considering  their  relation  to  the  ratio 
between  the  third  term  and  the  term  sought. 


CUMiUU^L>   riiOPORTION.  313 

Ki  i.i:. — Solve  by  successive  simple  proportions^  introducing  tJie 
conditions  one  at  a  time.     Or, 

Use  for  the  third  term  Vie  number  which  is  of  the  same  kind 
as  die  term  required. 

Arrange  the  like  numbers  in  couplets,  as  in  simple  proportion. 

The  product  of  the  means  divided  by  the  product  of  tJie  ex- 
tremes will  be  the  term  required. 

Problems  in  compound  proportion  are  readily  solved  by  cause  and 
effect.     Example  1,  stated  by  cause  and  effect,  is  as  follows : 

1^  Cause.        2d  Cause.  1st  Effect.  2d  Effect. 

6  men    "^        7  men    ")         T  C 

2  days    [  :   {)  days    J-   :  :  •<  24  acres  :    <  50  acres 


men    ")         C 

days     r   :  :  -j  ! 

hours  3  (. 


^  uays     (   •    w  ""y»     r    •  '   -s  ^'t  acres  :     a 
10  hours  3      12  hours  3  (.  (. 

Means      G  X  2  X  10  X  56 


Extremes      7  X  12  X  24 


=  3J 


2.  If  15  men  can  dig  a  ditch  in  45  days  by  working  10 
hours  a  day,  how  many  days  will  it  take  20  men  to  dig  it, 
by  working  12  hours  a  day? 

3.  If  a  block  of  granite  6  feet  long,  3  feet  wide,  and  2  feet 
thick,  weighs  5940  j)ounds,  what  will  be  the  weight  of  a 
block  of  the  same  kind,  which  is  9  feet  long,  4  feet  wide 
and  3  feet  thick? 

4.  If  I  place  $1500  at  interest  for  18  months  and  receive 
81'>5  interest,  what  sum  must  I  place  at  interest  at  the  same 
nito,  so  that  I  may  receive  $275  interest  in  8  months  ? 

5.  If  it  cost  $180  to  support  5  grown  persons  and  3 
children  for  3  weeks,  what  will  it  cost  to  support  8  grown 
persons  and  G  children  for  7  weeks,  allowing  that  it  costs  \ 
as  much  to  support  a  child  as  a  grown  i)erson? 

6.  If  20  men  working  8  hours  a  day,  can  dig  a  trench 
65  feet  long,  9  feet  wide  (ind  6  feet  deep,  in  25  days,  how 
many  days  will  it  take  25  men,  working  10  hours  a  day,  to 
dig  a  trench  75  feet  long,  8  feet  wide,  and  7  feet  deep? 


314  rK()iM)i:ri()N. 

^    7.  If   it  costs  $240   to   board    IG    ix3r&<>ijri    5  weeks,  how 
iniK  h  will  it  cost  to  board  9  persons  22  weeks? 

8.  If  $800  placed  at  interest,  amounts  to  $880  in  15 
months,  what  sum  must  be  placed  at  interest  at  the  same 
rate,  to  amount  to  $975  in  one  year? 

9.  If  it  requires  275  yards  of  cloth  f  yd.  wide  to  make 
75  garments,  how  many  yards  of  cloth  IJ  yd.  wide,  will  it 
require  to  make  215  such  garments? 

1<».  If  a  bin  which  is  8  feet  long,  6  feet  wide  and  8  feet 
iK  r|»,  holds  309  bushels  of  wheat,  how  inany  bushels  will  a 
l»iii  li«»ld  that  is  14  feet  long,  8  fcri  \\i  1.   and  9  feet  deep? 

11.  If  15  men,  working  10  hours  a  day,  can  do  a  certain 
piece  of  work  in  18  days,  how  many  days  will  it  rrquiro  for 
13  men  to  do  the  same  work,  by  working  -^  Imurs  ;i  day? 

12.  If  12  horses  consume  40  bushels  of  oats  in  8  days, 
how  long  will  140  bushels  of  oats  last  IH  horses? 

13.  If  a  regiment  of  1025  soldiers  c  ii>uni<  1  MOO  pounds 
of  bread  in  15  days,  how  many  pounds  will  3  regiments 
of  the  same  size,  ronsumo  in  12  days? 

14.  If  tlu'  wain-  that  fills  a  vat,  which  is  8  feet  long, 
4  feet  wide  and  5  feet  deep,  weighs  10000  pounds,  what  will 
be  the  weight  of  the  water  required  to  fill  a  vat,  which  is 
10  feet  long,  5  feet  wide  and  6  feet  deep? 

15.  If  5  horses  eat  as  much  as  6  cattle,  and  8  horses  and 
12  cattle  eat  12  tons  of  hay  in  40  days,  how  much  hay  will 
be  needed  to  kec])  7  horses  and  15  cattle  6o  days? 

16.  If  15  men  working  6  hours  a  day,  can  dig  a  cellar 
80  feet  long,  60  feet  wide  and  10  feet  deep  in  25  days,  how 
many  days  will  it  require  25  men  working  8  hours  a  day,  to 
dig  a  cellar  120  feet  long,  70  feet  wide  and  8  feet  deep? 

17.  If  52  men  can  dig  a  trenr.h  355  feet  long,  60  feet 
wide  and  8  feet  deep  in  15 -days,  how  long  will  a  trench  be, 
that  is  45  feet  wide  and  10  feet  deep,  \vhich  45  men  can  dig 
in  25  days? 


409.  1.  Of  what  number  are  3  and  3  the  factors?  4 
and  4? 

2.  Of  what  number  are  3  and  3  and  3  the  factors?  4  and 
4  and  4? 

3.  What  is  the  product  when  5  is  used  twice  as  a  factor? 

4.  What  is  the  product  or  power,  when  6  is  used  twice  as 
a  factor?    When  8  is  used  twice  as  a  factor? 

5.  What  is  the  product  of  |  X  J  ?     Of  }  X  f  ? 

6.  What  is  the  product  when  ^  is  used  twice  as  a  factor  ? 
When  "I  is  used  three  times  as  a  factor? 

7.  What  is  the  product  of  two  4's,  or  the  second  power 
of  4?  What  is  the  product  of  three  5*s,  or  the  third  pmcer 
of  5?    What  is  the  third  power  of  6? 

8.  What  is  the  second  power  of  |?    Of  }?    Of  |? 


DEFINITIONS. 

500.  A  Powei*  of  a  number  is  the  product  arising  from 
using  the  number  a  certain  number  of  times  as  a  factor. 

501.  The  powers  of  a  number  are  named  from  the  number 
of  times  the  number  is  used  as  a  factor. 

Thus,  when  2  is  used  on  a  factor  twice,  the  product,  4,  is  called  the 
Kcond  pcnctT  of  2.  9  is  the  tccond  power  of  3.  27  is  the  third  powtr 
of  3. 

The  number  itself  b  called  the  lint  power. 

(S15) 


316  INVOLUTION. 

602.  The  number  of  times  a  number  is  used  as  a  factor  is 
indicated  by  a  small  figure  called  an  Exponent^  written  a 
little  above  and  at  the  right  of  the  number. 

ThiiB,  3^  means  the  aeeond  pmcer  of  3;  5*,  the  fourth  poicer  of  "),  vtc. 

Inasmuch  aa  the  area  of  a  square  is  the  product  of  tuo  equal  faoton, 
and  the  volume  of  a  cube  is  the  product  of  three  equal  faclors,  the  second 
power  of  a  number  is  also  called  the  square^  and  the  third  power  the 
cube  of  the  number, 

503.  Involution  is  the  process  of  finding  the  power  of 
a  number. 

wm  I  I  I  N     /   \rnrTSES, 

504.  1.  Find  the  third  power  of  15. 

PROCESS.  Analysis. — To    tuul    tlie    tliinl 

icv^-itcvyiK        ooTK        power  of  a  number  is  to  find  the 

product,  when  the  number  is  used  3 
times  as  a  factor.  Therefore,  the  third  power  of  15  will  be  15  X  15 
X 15,  which  is  equal  to  3375. 

2.  Find  the  third  power  of  12.     i  :      39.     24. 

3.  Find  the  second  power  of  47.     51.     29.     34. 

4.  What  is  the  square  of  15?     33?    24?    36?    25? 

5.  What  is  the  cube  of  28?    45?     18?     21?     41? 

6.  What  is  the  third  power  of  f  ?     A  ns.  -^  X  -f  X  -f  =  HI- 

7.  What  is  the  cube  of  I?     Of|?     f?    A?    tV? 

8.  What  is  the  fourth  power  of  f  ?     Cube  of  ^V  ^ 

Find  the  value  of  the  following : 


9.  15*. 

12. 

.05*. 

15. 

(HY- 

18. 

(25i)'. 

10.  25'. 

13. 

.005'. 

16. 

(W- 

19. 

(3,00^)'. 

11.  30'. 

14. 

2.05\ 

17. 

(41)'. 

20. 

(4.5001)'. 

21.  Raise  10  to  the  fourth  power;   8  to  tlio  third  power; 
3  to  the  6th  power. 


INVOLUTION. 


317 


PROCESS. 

•J -3 

35 

25  = 

u' 

;i} 

=  2tXu 

9 

=  t' 

1225  = 

t'  +  2tX 

«  + 

«' 

*505.  To  find  the  Hqtiure  or  a  nnmber  in  terms  of*  its 
partM. 

1.  Find  the  square  of  35  in  terms  of  its  tens  and  units. 

Analysis. — If  we  square  35  or 
multiply  35  by  itself  and  write 
every  step  in  the  process,  we  shall 
have  for  the  first  product  25,  or  the 
8(]uare  of  the  units,  for  the  next  two 
products  15  tens,  or  two  times  the 
product  of  the  tens  and  units,  and 
for  the  third  product  9  hundreds  or 
the  square  of  the  tens.     Hence, 

506.  Principle. — The  square  of  any  number  consisting  of 
tens  and  unitSj  is  equal  to  the  tens*  -f  2  times  the  tens  X  tJie  units 
-f  the  unit^. 

Thus,  25  =  20  +  5,  and  25^  =  20^  +  2  (20  X5)  +  5  2. 

The  above  principle  is  true  into  whatever  two  parts  the  number 
may  be  separated,  and  the  principle  stated  in  general  terms  wonlil 
be,  the  square  of  any  number  consisting  of  two  parts  is  equal  to  the 
first  part  2  -f  2  times  the  first  part  X  the  second  -|-  second  part'. 

Thus,  14  =  8  +  6,  and  142  =  82  +  2  (6X8) +  C2. 

Express  in  terms  of  their  ton?  and  units  the  square  of  the 
following  numbers: 


2.  64. 

5.  47. 

8.  74. 

11.  39, 

;j.  71. 

6.  80. 

9.  95. 

12.  44. 

4.  68. 

7.  2G. 

10.  82. 

13.  67 

1  1.  Square  16  by  squaring  its  parts    9  and  7. 

1 ").  Square  20  by  squaring  its  parts  12  and  8. 

11).  Square  32  by  squaring  its  jMirts  30  and  2. 

17.  S(|uare  13  by  squaring  it«  parts    7  and  6. 

18.  Sfjuare  26  by  squaring  its  jwrts    9  and  17, 

19.  SquH'-"  17  bv  <..i„iring  its  parts    8  and  9. 


318 


INVOLUTION. 


507.   To  And  Uie  cabe  of  a  Dumber  in  terms  or  its 
partN. 

1.  Find  the  cube  of  36  in  terms  of  its  parts. 

PBOCESS. 

126  =  u» 
75 


36»  = 


25 
15 
15 
9 


X35  = 


75 
75 
45 
45 
45 
27 


}- 
}- 


3«Xti' 


3<«Xt« 


=  f 


42875  =  <»4  3^  X  u  4- 3e  X  w«  +  «* 

Analysis. — By  multiplying  the  second  power  expressed  as  in 
Art.  ii05,  by  35,  and  writing  every  step,  we  shall  have  the  cube  of 
the  tens,  plus  the  product  of  three  times  the  square  of  the  tens  multi- 
plied by  the  units,  plus  the  product  of  three  times  the  tens  multiplied 
by  the  square  of  the  units,  plus  the  cube  of  the  units.    Hence, 

508.  Principle. — The  cube  of  any  number  consisting  of  tens 
and  unii8  is  equal  to  the  tens  *  -f-  3  times  the  tens  '  X  the  units 
-f  3  times  the  tens  X  the  units^  -f-  the  uniis^. 

Thus,  25  =  20  +  5,  and  25' =  20' +  3(20^  X  5)  +  3  (20X  5^)  +  5'.     I 

The  above  principle  may  be  stated  in  general  terms  thus:  The  cube 
of  any  number  when  separated  into  two  parts  is  equal  to  the  first  part  ^ 
-f  3  times  the  first  part  -  X  second  part  -f  3  times  the  first  part  multi- 
plied by  the  second  part  -  +  the  second  part  ^ 

Express  in  terms  of  their  tens  and  units  the  cube  of  the 
following  numbers: 


2.  26. 

5.  42. 

8.  38. 

11.  52. 

3.  31. 

6.  27. 

9.  39. 

12.  64. 

4.  28. 

7.  36. 

10.  54. 

13.  66. 

509.  1.  What  are  the  factors  of  36?  What  are  the  two 
equal  factors  of  36  ?     Of  49  ?    Of  81  ? 

2.  What  number  used  three  times  as  a  factor  will  produce 
27?    64?     125?    216? 

DEFINITIONS. 

510.  A  Root  of  a  number  is  one  of  the  equal  factors  of 
the  number. 

Thus,  4  is  a  root  of  16,  because  it  is  one  of  two  equal  factors. 

Roots  are  named  in  a  manner  similar  to  powers.  Thus, 
one  of  tivo  equal  factors  of  a  number  is  the  second,  or  square 
root;  one  of  three  equal  factors,  the  third,  or  cube  root;  one 
of  four  equal  factors,  the  fourth  root,  etc. 

611.  ^Evolution  is  the  process  of  finding  roots  of  num- 
bers. 

512.  The  liadivul,  or  Root  Siffn,  is  y/.  When 
placed  before  a  number  it  shows  that  its  root  is  to  be  found. 

When  no  fijjure  or  iiuiex  is  written  in  the  opening  of  the 
radical  sign,  the  square  root  is  indicated;  if  the  figure  3  is 
placed  there,  as  ^,  the  cube  root  is  indicated;  if  4,  as  y  , 
the  fourth  root;  etc. 

513.  A  Perfect  Power  \a  a  number  whose  root  can  be 
found. 


320  K  VOLUTION. 

514.  An  Itnitrrj'rri  Vowcv  \&  a  number  whose   rout 
can  not  be  fouinl  •  xm  nv. 


EVOLUTION  BY  FACTORING. 

515.    1.  What  is  the  8qi>are  root  of  1225? 

PROCtSB.  Analysis. — Since  the 

5)1225  square  root  of  a  number 

is  one  of    its  two  equal 
factors,  we  may  iind  the 


5)245 


7)49  square  root  of    1225   by 

7)7     l/i225  =  5  X  7  =  35       separating     it     into     its 

prime  factors,  and  find- 
ing the  product  of  the  numbers  forming  one  of  the  two  equal  sets  of 
factors.  The  prime  factors  are  5,  5,  7  and  7,  and  5  and  7  form  one  of 
the  tVo  equal  '^ets.  Therefore  their  product,  35,  is  the  square  root 
of  1225. 

Rule. — ^Separate  fh/b  numbers  into  their  prime  factors.  Ar- 
range these  factors  into  twOj  three^  four,  or  any  number  of  sets 
containhig  the  sam^efadorSf  according  as  the  second,  third,  fourth, 
or  any  root  is  to  be  found, 

Tlie  product  of  Hie  fa/Aors  which  form  a  set  wUl  be  the  root. 

This  method  is  valuable  only  when  the  numbers  whose  roots  are 
Bought  are  perfect  powers. 

2.  Find  the  square  root  of    144.     256.       324.     576. 

3.  Find  the  cube    root  of     64.     512.     4096.     13824. 

4.  Find  the  fourth  root  of  1296.     The  fifth  root  of  248832. 

Orders  of  Units  vs  Powers  and  Roots. 

516.  1.  How  does  the  number  of  figures  which  express  the 
square  of  units,  compare  with  the  number  expressing  units  ? 

2.  How  does  the  number  of  figures  required  to  express  the 
cube  of  units  compare  with  the  number  expressing  units? 


EVOLUTION.  321 

3.  Write  the  numl)ers,  10,  99,  100,  999,  1000,  and  under 
tlieni  tlu'ir  second  and  third  powers. 

4.  How  does  the  number  of*  figures  required  to  express  the 
p(>cond  powers  compare  witii  the  number  of  figures  required 
!  >  express  the  given  numbers? 

5.  How  does  the  number  of  figures  required  to  express  the 
third  powers  compare  with  the  number  of  figures  required  to 

repress  the  given  numbers? 

517.  Principles. — 1.  TJie  square  of  a  number  is  expressed 
by  twice  as  many  figures  as  the  number  itself  or  one  less. 

2.  Tlie  cube  of  a  number  is  expressed  by  three  times  as  mxiny 
fig^ires  as  Hie  number  itself  or  one  or  two  less. 


EXERCiaES. 

518.  Tell  by  referring  to  the  principles,  how  many  figures 
tlierc  are  in  the  following: 

1.  In  the  square  of  21.  Of  15.  Of  115.  Of    4156. 

J.  In  the  cube     of  19.  Of  25.  Of  316.  Of    6184. 

">.   In  the  square  of  35.  Of  29.  Of  584.  Of    8196. 

4.  In  the  cube     of  59.  Of  67.  Of  999.  Of    9999. 

T),  How  many  figures  or  orders  of  units  are  there  in  a  num- 
lx>r  if  the  second  power  of  it  is  expressed  by  4  figures?  By  7 
figures?     By  9  figures? 

6.  How  many  figures  or  orderfe  of  units  are  there  in  a  num- 
\niT  if  the  third  power  of  it  is  expressed  by  6  figures  ?  By  8 
figures?  By  12  figures?  By  11  figurte?  By  21  figures? 
By  25  figures? 

519.  Principlhs. — 1.  The  orders  of  units  in  the  square  root 
■fa  mtmlHr  corrvi*}Hmd  to  tJie  number  of  periods  of  two  figures  each 
into  which  the  numbt^  can  be  separated,  bcoinninq  at  units. 

21 


322 


EVOLUTION. 


2.  The  orders  of  units  in  the  cube  root  of  a  number  correspond 
to  the  number  of  periods  of  tiiree  figures  each  into  which  Uie  number 
can  be  separated,  beginning  at  units. 


SQUARE  ROOT. 

520.    1.  What  is  the  square  root  of  576,  or  what  is  the 
side  of  a  square  whose  area  is  576  square  units  ? 


1st  Procebs. 

576(20 


20*  =  400 

2X20  =  40)176 
(-10   '    n  x4  =  176 


m 


:z:±^t*MrAn 


s 


Analysis. — According  to  Prin. 
1,  Art.  aiOf  the  orders  of  unita 
in  the  stjuare  root  of  any  number 
may  be  determined  by  separating 
the  number  into  periods  of  two 
figures  each,  beginning  at  units. 
Separating  57G  thus,  there  are 
found  to  be  two  orders  of  units  in 
tlie  root,  or  it  is  composed  of  tens 
and  units.  Since  the  square  of 
tens  is  hundreds,  5  hundreds  must 
be  the  square  of  at  least  2  tens.  2 
tens  or  20  s(juared  is  400,  and  400 
subtracted  from  576  leaves  17G, 
therefore  the  root  20  must  be  in- 
creased by  such  an  amount  as  will 
exhaust  the  remainder. 

The  square  (A)  already  formed 
from  the  576  square  units  is  one 
whose  side  is  20  units,  but  inas- 
much as  the  number  of  units  was 
not  exhausted,  such  additions  must 
be  made  to  the  square  that  they 
will  exhaust  the  units  and  keep  the 
figure  a  square.  The  necessary  ad- 
ditions are  two  equal  rectangles 
B  and  C,  and  a  small  square  D. 
Since  the  square  D  is  small,  the  area  of  the  rectangles  B  and  C 


SQUARE   ROOT.  323 

is  nearly  ITt*  units.  The  area,  176  units,  divided  by  the  length  of 
♦ho  rectangles,  will  give  the  width,  which  is  4  units.  The  width  of 
'ho  additions  is  4  units,  and  the  entire  length,  including  the  small 
«*<iuare,  is  44  units;  therefore  the  area  of  all  the  additions  is  4  times 
*4  units,  or  176  square  units,  which  is  equal  to  the  entire  number 
o(  units  to  be  added.  Therefore  the  side  of  the  square  is  24  units, 
or  the  square  root  of  the  number  is  24. 

2d  process.  Analysis.— In  the  same  manner 

5*7  6  (24     *^  before,  the  root  of  this  number  is 

.J n  J A  shown  to  consist  of  tens  and  units. 


2e  =  40 
u==    4 

2«  +  t<  =  47 


The  tens  can  not  be  greater  than 
1  '  ^  2;  therefore  we  write  2  tens  for  a 

partial   root.     Squaring    and   sub- 
17  6  tracting,   there  is    a   remainder  of 

176,  which  is  composed  of  2  times 
the  tens  X  "nits  +  units^,  Art.  506,  Since  2  times  the  lens  multi- 
plied by  the  units  is  much  greater  than  the  units  squaretl,  176  is  nearly 
two  times  the  tens  multiplied  by  the  units.  Therefore  if  176  is  di- 
vided by  twice  the  tens,  or  40,  the  quotient  will  be  approximately 
the  units  of  the  root.     Dividing,  the  units  are  found  to  be  4. 

Since  the  tens  are  to  be  multiplied  by  the  units,  and  the  units  are 
to  be  multiplied  by  the  units  or  squared,  and  these  results  are  to  be 
added,  to  abridge  the  process  the  units  are  added  to  twice  the  tens  and 
the  sum  multiplied  by  the  units.  Thus,  40  +  4  is  multiplied  by  4, 
making  176.    Therefore  the  square  root  of  576  is  24. 

When  the  root  consists  of  more  than  two  orders  of  units, 
the  process  for  solving  is  similar  to  that  already  given. 

2.  Find  the  square  root  of  137641. 

IST  PROCESS.  2d  process. 

13-76-41  (300  13-76-41(371 

9  00  00     70  9 

300x2:=600-f  70=670)4  76  41  1  67)476 

4  69  00  371  469 


370x2=740+1=741  )7  41  741)741 

7  41  741 


324 


jAoi.r  rioN. 


In  tlic  first  process  the  stoD?  arc  dv*  n  ^^  i; 
fullness,  while  in  the  S( 
as  much  as  |)o>>il»K'. 


•iisiderable 
abbreviated 


Rule. — Separde  the  number  into  periods  of  two  figures  eachf 
heqlniunq  at  unit.'*. 

ff(:.'<t  !<fjufirc  in  tfie  left-Hiand  periody  a///  "  /  /    'V>" 
root  Jar  the  Just  figure  of  the  required  root. 

Square  thi.<   mnf  mid  s-uhfrrirf  the  ra^ult  from  the  left-hand 
periody  ami  the  next  period  for  a  dividend. 

Double  the  r  j  jomtii  for  a  trial  dicisory  and  by  it 

divide   tJie  divi  ■  qnrdluf]  tlie  right-hand  figure.      The 

quotient  or  quoti  be  Vie  second  figure  of  the  root. 

Annex  to  the  tri'U  (un-  complete  divisor,  the  figure  last 

found,  mvlfiphj  fhi^  div'  la.ft  figfir^  of  the  root  found, 

■nil'  r  (I  II II'  x 


thii.<. 


■  .I'd  nil  the  periods  liave  been  used 
it  will  be  the  square  root  sought. 


1.  Wiuii    till'    iiuin1)rr   is    li'.t   a    j>-rJ\i- 


.!.■  X    ii>  riods   of 
•J.    I  -h.  by 

l)e.MlUiiiiy^  III    U'lu  : 

.'>.  The  square  nil    l»y  extracting 

the  squar*.'  rodt   ol    b( ■ili   liiiiiiLiMi  .r   a-al   denominator  scparatL-ly,  or 
by  reducing  it  to  a  dccinial  and  then  extracting  its  root. 

Extract  the  square  root  of  the  following : 


3.  2809. 

7.  7* '756. 

11. 

938961. 

4.  3969. 

8.  118336. 

12. 

5875776. 

5.  4356. 

9.  674041. 

13. 

12574116. 

6.  9216. 

10.  784996. 

14. 

30858025. 

15.  Find  the  vahic  of  ]   222764;  i  11390625. 


16.  Find  tbe  value  of  i   .763876;  i/.30&58025. 


SQUARE   ROOT.  325 


17.  What  i3  the  square  root  of  .098G86? 

18.  What  is  tlie  stjuare  root  of  .099225? 

19.  What  is  the  square  root  of  WWoT^ 

20.  Extract  the  square  root  of  ^^ff • 

21.  Extract  the  square  root  of  f  |H?4- 

22.  Extract  the  square  root  of  J. 

23.  Extract  the  square  root  of  j. 

24.  Extract  the  square  root  of  f . 

25.  Extract  the  square  root  of  .9. 


APPLICATIONS  OF  SQUARE  ROOT. 

5*21.  To  flfid  the  sl<le  or  a  sqnaro  when  its  aroa  is 
Riven. 

Since  tlie  area  of  a  square  is  the  product  of  two  ccjual 
factors  which  represent  its  sides,  the  sides  may  be  found 
by  extracting  the  square  root  of  the  number  expressing  its 

1.  What  is  I  Ik  side  of  a  square  whose  area  is  625  square 
feet? 

2.  What  is  the  side  of  a  square  whose  area  is  2025  square 
rods? 

3.  A  rectangle  whose  area  is  5408  square  feet  is  composed 
of  two  equal  squares.     What  is  the  length  of  its  sides? 

4.  A  man  owns  50  acres  of  land  in  two  square  fields,  one 
of  which  contains  4  times  as  much  area  as  the  other.  How 
many  nxls  of  fence  will  be  needed  to  fence  both  fields  if 
they  arc  not  adjacent? 

5.  Tlie  letij^th  of  a  rectangular  field  containing  20  acres  is 
twice  its  width.     What  is  the  distance  around  it? 

6.  If  it  cost  8572  to  inclose  with  a  fence  a  field  that  is 
72  n)ds  long  and  32  nnls  wide,  how  much  less  will  l)c  the 
cost  (if  iiu'losinir  :i  s<juinv  field  contaiiiiiiL'"  the  same  urea? 


326 


EVOLUTION. 


522.   To  find  any  nide  of  a  right«iigled  triangle  wh«n 
the  other  MideM  are  {^iven. 


Triangle 


Bight  aoKl^ 


523.  A  Tniuiif/le  is  a  figure  which    has  three   angles 
and  three  sides. 

524.  A  jRiyht  Anf/le  is  the  angle  formed  when  one 
line  is  drawn  perpendicular  to  another. 

525.  A  Right -angled  Triangle  is  a  triangle  which 
has  a  right  angle. 

526.  The  Hypotenuse  of  a  right-angled  triangle  is  the 
side  opposite  the  right  angle. 

527.  The  Sase  of  a  triangle  is  the  side  on  which  it  is 
assumed  to  stand. 

528.  The  Perjyenilicular  is  the  side  which  forms  a 
right-angle  with  the  base. 

The  relation  of  the  squares  de- 
scribed upon  the  sides  of  a  right- 
angled  triangle  is  expressed  thus: 

529.  Principles.  — 1.  The 
square  described  upon  the  hypote- 
nuse of  a  right-angled  triangle  is 
equal  to  the  stim  of  Hie  squares  on 
the  other  two  sides. 

2.  The  square  on  either  of  the 
other  sides  of  a  right-angled  triangle  is  equal  to  the  square  on  the 
hypotenuse  diminished  by  the  square  on  the  other  side. 


4 


SQUARE   ROOT.  327 

"When  the  numl)er  of  square  units  in  the  surface  of  any  square 
figure  is  known,  its  side  may  be  found  by  extracting  the  square 
root  of  the  number  according  to  the  preceding  case. 

530.  1.  The  base  of  a  right-angled  triangle  is  8  feet  and  the 
perpendicular  6  feet,  what  is  the  hypotenuse  ? 

PROCESS.  Analysis. — Before  we  can  determine  the 

at    '   at       A  f\c\  length  of  the  hypotenuse  when  the  sides  are 

"    ~T-  "         J  U  U  .         given  we  must  find  the  area  of  a  wjuare  de- 

j/  "JlOO  =:  1  0 .  scribed  upon  it.     The  square  described  upon 

it  is  equal  to  the  sum  of  the  squares  upon 

the  other  two  fiides,  or  the  sum  of  8^  +  6-,  which  is  100.    Since  the  area 

of  tlic  square  described  upon  the  hypotenuse  contains  100  square  units, 

the  length  of  the  side  is  the  square  root  of  100  or  10. 

^o  find  the  liypotcnnse. 

Rule. — Extract  the  square  root  of  the  sum  oj  the  squares  of  the 
other  two  sides. 

To  fliKi  the  base  or  perpeiidicnlar. 

Rule. — Extract  the  square  root  of  the  difference  of  the  squares 
on  Vie  hypotenuse  and  the  other  side, 

2.  The  base  of  a  right-angled  triangle  is  15  feet  and  the  per- 
pendicular is  20  feet.     What  is  the  hypotenuse  ? 

3.  Tlie  base  of  a  right-angled  triangle  is  40  feet  and  the 
hyi)otenuse  is  120  feet.     What  is  the  perpendicular? 

4.  The  perpendicular  of  a  right-angled  triangle  ia  30  feel 
and  the  hyjwtenuse  is  50  feet.     What  is  the  base? 

5.  A  tree  150  feet  high,  standing  upon  the  bank  of  a  stream, 
was  broken  off  125  fi^t  from  the  top,  and  falling  across  the 
stream  the  top  just  reached  the  otlur  sliurc  What  was  the 
width  of  the  stream  ? 

6.  Two  steamers  start  from  tlic  >ame  point,  one  g<»ing  due 
north  at  the  rat(^  of  15  miles  an  hour,  and  the  other  going  due 


328  EVOLUTION. 

west  at  the  rate  of  18  miles  an  hour.    How  iar  apart  were  they 
at  the  end  of  6  hours  ? 

7.  A  rectangular  park,  whose  sides  are  respectively  45  rods 
and  60  rods  in  length  litus  a  walk  crossing  it  from  corner  to 
corner.     How  long  is  the  walk  ? 

8.  A  certain  assembly  room  is  100  feet  in  Icngtli,  GO  feet  in 
width,  and  26  feet  in  height.  What  is  the  distance  from  a 
lower  comer  to  the  upper  opposite  comer  ? 

9.  Two  buildings  standing  opposite  each  other  are  respect- 
ively 60  feet  and  80  feet  liigh.  A  ladder  125  feet  long  placed 
at  a  certain  distance  from  the  base  of  each  just  reaches  the 
top  of  each.     How  far  apart  are  the  buildings? 

10.  The  distance  from  the  base  of  a  building  to  a  pole  is 
145  feet,  and  a  string  225  feet  long  attached  to  the  top  of  the 
pole  just  reaches  the  base  of  the  building.  What  is  the  height 
of  the  pole  ? 

11.  A  person  wlio  wished  to  ascertain  the  exact  height  of 
St.  Paul's  Cathedral  in  London,  England,  learned  by  inquiry 
that  a  rope  extending  from  the  top  of  the  cross  to  a  point 
300  feet  from  tlie  center  of  the  circular  pavement  under  the 
dome  was  488  feet  104-  inches  long.  If  the?e  data  were  cor- 
rect, what  is  the  height  of  St.  Paul's '/ 


SIMILAR  FIGURES. 

531.  Similar  Figures  are  such  as  are  of  the  same 
form,  and  differ  from  each  other  only  in  size. 

The  truth   of  the   following  principles  can  be  shown  by 
geometry : 

532.  Principles. — 1.  Similar  surfaces  are  to  each  oilier  as 
the  squares  of  their  corresponding  dimensions.     Hence, 

2.    The  correspo)iding  dimensiojis  of  similar  sntfa^es  are  to  each 
other  as  Hie  square  roots  of  iJieir  areas. 


siiiAKi:  HOOT.  329 

1.  If  the  area  of  a  triangle  whose  base  is  16  rods,  is  128 
square  rods,  how  many  square  rods  are  there  in  the  area  of  a 
similar  triangle  whose  base  is  12  rods  ? 

PROCESS.  Analysis. — Since  the  areas  of  similar 

1A2      10»  figures  are  to  each  other  as  the  squares 

"     *      *  *  *  'of  their  like  dimensions,  the  area  of  the 

128  :  X  : :  256  :  144  first  triangle  (128  sq.  rd.)  will  be  to  the 

X:=72aa    rd  ^^^^  ^^  *'^^  second  triangle  (x)  as  the 

square  of  the  side  of  the  first  triangle 

(16^)  is  to  the  square  of  the  side  of  the  second  triangle  (122).   Solving 

the  proportion,  the  area  is  72  sq.  rd. 

2.  If  the  area  of  a  circle,  whose  diameter  is  2  feet,  is  6.2832 
sq.  ft.  what  will  be  the  area  of  a  circle  whose  diameter  is  4  feet? 

3.  If  the  side  of  a  rectangular  field  containing  25  acres  is 
40  rods,  what  is  the  side  of  a  similar  field  containing  10  acres? 

Analysis.— vSince  the  corresponding  dimensions  of  similar  surfaces 
arc  to  each  other  as  the  square  roots  of  their  areas: 

v/25  :  v/iO  ::  40  :  x,  or  5  :  /lO  ::  40  :  x. 

Extracting  the  square  root  of  10  and  solving  the  proportion,  x,  or 
the  corresponding  side,  is  25.296  rd. 

4.  If  the  side  of  a  square  field  containing  40  acres  is  80 
rods,  wlinf  will  be  the  side  of  n  similar  field  whose  area  is  25 
acnv- 

5.  It  the  urea  of  a  circle  who::e  diunieter  is  20  feet  is  314.16 
s<|uare  feet,  what  is  the  diameter  of  a  circle  whose  area  is 
113.0976  square  feet? 

6.  A  farmer  has  two  rectangular  fields  simihir  in  lorn  : 
one,  whose  length  is  120  roils  and  whose  breadth  is  12  rods, 
contains  9  acres,  the  other  contains  GJ  acres.  What  are  its 
length  and  breadth? 

7.  A  horse  tied  to  a  stake  by  a  rope  7.13  rods  long  can 
graze  uiK)n  just  1  acre  of  ground.  How  long  must  the  rt>pe 
be  that  he  may  graze  upon  5  acres? 


330 


EVOLUTION. 


CUBE   ROOT 


1st  process. 


20»  = 
3  X  20«  =  1200 
3  X  4  X  20  =  240 
4«=     16 


1.3-824(20-1-4=24 
8000 


1456 


5  824 


5  824 


633.    1.  What  is  the  cube  root  of  13824,  or  what  is  the 
edge  of  a  cube  whose  solid  contents  are  13824  units? 

Analysis. — Accord- 
ing to  Pr.  2,  Art.  5  i  9, 
the  orders  of  units  in 
the  cube  root  of  any 
number  may  be  de- 
termined from  the 
number  of  periods  ob- 
tained by  separating 
the  number  into  pe- 
riods, containing  three 
figures  each,  beginning  at  units.  Separating  the  given  number  thus, 
there  are  two  periods,  or  the  root  is  composed  of  tens  and  units. 

The  tens  in  the  cube  root  of 
the  number  can  not  be  greater 
than  2,  for  the  cube  of  3  tens  is 
27000.  2  tens,  or  20  cubefl,  are 
8000,  which,  subtracted  from 
13824,  leave  5824;  therefore 
the  root,  20,  must  be  increased 
by  a  number  such  that  the 
additions  will  exhaust  the  re- 
mainder. 

The  cube  (A)  already  formed 
from  the  13824  cubic  units  is 
one  whose  edge  is  20  units. 
The  additions  to  be  made, 
keeping  the  figure  formed  a 
perfect  cube,  are  3  equal  rect- 
angular solids,  B,  C  and  D; 
3  other  equal  rectangular 
solids,  E,  F  and  G;  and  a 
small  cube,  H.  Inasmuch  as 
the  solids,  B,  C  and  D,  com- 


CUBE    ROOT. 


331 


prise  much  the  greatest  part  of  the  additions,  their  solid  contents 
will  be  nearly  5824  cubic  units,  the  contents  of  the  addition. 

Since  the  cubical  contents 
of  these  three  equal  solids  are 
nearly  equal  to  5824  units,  and 
tlie  superficial  contents  of  a 
-ide  of  each  of  these  solids  are 
iV)  X  20,  or  400  square  units, 
if  we  divide  5824  by  3  times 
400,  or  1200,  since  there  are  3 
equal  solids,  we  shall  obtain 
the  thickness  of  the  addition, 
which  is  4  units. 

Since  all  the  additions  have 
the  same  thickness,  if  their  su- 
IK'rficial  contents,  or  area  of 
fiich  side,  are  multiplied  by 
4,  the  result  will  be  the  solid 
contents  of  these  additions. 

Iksides  the  larger  additions 
there  arc  three  others,  E,  F, 
and  G,  that  are  each  20  units 
ling  and  4  units  wide,  or  whose 
ides  have  an  area  of  80  units  each,  and  {he  area  of  all  240  units,  and 
a  small  cube  whose  .-'vies  have  an  area  of  16  units.  The  sum  of  these 
areas,  1456,  multiplied  by  4,  the  thickness  of  the  additions,  gives  the 
solid  contents  of  the  additions,  which  are  5824  units. 

Therefore  the  edge  of  the  cube  is  24  units  in  length,  or  the  cube 
root  of  13824  is  24. 


2d  PBOCE88. 

3i«  =  20*x3  =  1200 

3«Xtt=(20x4)x3=   240 

n«=;4x4=     16 


13.824(24 
8  000 


3«»-f  3/u4-M«  =  1456 
(3«*4-3«u-i-u«)Xti=: 


6  824 


5  824 


Analysis. — In 
the  same  manner 
as  before,  it  may 
be  shown  that 
the  root  of  the 
nuniWr  contains 
only  tens  and 
units.  The  lens 
can  not  Ihj  great- 
er than  2,  for  3 


332 


EVOI^UTION. 


tens  cubed  would  equal  27000.  Cubing  and  eubtracting,  tlicre  18  left 
6824,  which  in  composed  of  three  timeu  the  tens^  X  ^be  units  -f  3  times 
the  tens  X  the  units'-  +  the  unitir*,  Art.  507* 

Since  3  times  the  tens'*  is  much  greater  than  3  times  the  tens  X  th© 
units^  -j-  the  units'^  5824  is  a  little  more  than  3  times  the  tens^  X  the 
units.  If,  then,  5824  is  divided  by  3  times  the  tens^,  or  1200,  the  trial 
divisor,  the  quotient  4,  will  be  approximately  the  units  of  the  root. 

Since  5824  is  equal  to  the  sum  of  3  times  the  tens^  multiplied  by 
tlie  units,  3  times  the  tens  multiplied  by  the  units-  and  the  unitH-*^,  the 
process  may  be  shortened  by  adding  together  3  times  the  tens-,  3  times 
the  tens  X  the  units  and  the  units^,  and  multiplying  this  sum,  1456, 
by  the  units,  4.  The  product  is  5824,  which,  subtracted  from  the  num- 
ber, leaves  no  remainder. 

"When  the  root  consists  of  more  than  two  orders  of  units 
the  processes  and  analyses  are  similar  to  those  already  given. 


2.  What  is  the  cube  root  of  48228544? 


300>  = 
3X300*  =  270000 
3X300X60=  54000 
60"=     3600 
327600 
3x360^  =  388800 
3x360x4=     4320 

4'= 16 

393136 


48-228-544 
27  000  000 


^  228  544 


19  656  000 


300 

60 

4 

364 


1  572  544 


1  572  544 


Rule. — Separate  the  number  into  periods  of  three  fibres  mc^, 
beginning  at  units. 

Find  the  greatest  cube  in  the  left-hand  period,  and  write  its 
root  for  Hie  first  part  of  tJie  required  root. 

Cube  this  root,  subtract  tJie  result  from  the  left-hand  period, 
ami  awiex  to  the  remainder  the  next  period  for  a  dividend. 

Take  3  times  the  square  of  the  root  already  found  fm-  a  trial 


cLiij:  liuoT.  333 

din.-u, ,  aiKi  ''J  li  taV.ur  the  dividend.  The  quotient  or  the  quo- 
tient dimini'Jicd  will  be  the  second  'paH  of  the  root. 

To  this  tried  divisor  add  3  times  the  product  of  the  first  part 
of  the  root  by  Vie  second  part,  ami  also  the  square  of  tlic  second 
]Hirt.     Their  mm  will  be  tlie  entire  divisor. 

Multiply  die  entire  divisor  by  the  second  j)art  of  Vie  root  and 
-subtract  the  product  from  the  dividend. 

Continue  tlius  until  all  tJiefig^ires  of  the  root  have  been  found. 

1.  When  there  is  a  remainder,  after  subtracting  the  last  product 
annex  decimal  ciphers,  and  continue  the  process.  The  figures  of  the 
rcM)t  obtained  will  be  decimals. 

2.  Decimals  are  pointed  off  into  periods  of  three  figures  each,  by 
beginning  at  tenths  and  passing  to  the  right. 

3.  The  cube  root  of  a  common  fraction  is  foun4  by  extracting  the 
rube  root  of  both  numerator  and  denominator  separately,  or  by 
reducing  it  to  a  decimal  and  then  extracting  its  root. 


I'lxtract  the  cube  root  of  the  following: 

3.  7408H.         I       6.  704969. 

4.  2(;-J144.       1       7.   185193. 


liiiJ.JTr).  8.  25004 


9.  5545233. 

10.  2000376. 

11.  153990656. 


12.  What  is  the  cube  root  of  2  to  3  decimal  places? 

13.  What  is  the  culx)  root  of  9'to  4  decimal  jilarrs? 

14.  What  is  the  cube  root  of  .27?    Of  .64? 

1 5.  What  is  the  cube  root  of  |  ?    |  ?    TsUih^  ? 


APT^TJCWTIONS  OF  CUBE  ROOT. 

534.  1.  What  is  the  length  of  the  edge  of  a  cubical  box 
that  contains  91125  cubic  feet? 

J  What  are  the  dimensions  of  a  cubical  box  that  con- 
lainn  as  much  as  a  rectangular  box,  that  is  2  feet  8  inches 
long,  2  feet  3  inches  wide,  and  1  foot  4  inches  deep  ? 


334  EVOLUTION. 

3.  What  is  the  depth  of  a  cubical  cistern  whose  contents 
are  2197  cubic  K»et? 

4.  What  must  he  llic  drpth  ni  a  ciihiral  l)in  tluit  will 
contain  1000  bushels? 

5.  What  must  be  the  depth  of  a  cubical  cistern  that  will 
hold  300  barrels  of  water? 

6.  A  bin  that  ct)ntains  2000  bushels  of  grain,  is  just 
twice  as  long  aa  it  is  wide  or  high.     What  is  its  length? 

7.  What  is  the  depth  of  a  cubical  box  that  will  hold  a 
bushel  ? 

8.  What  is  the  depth  of  a  cubical  box  that  will  hold  a 
barrel  of  water  (Sl\  gal.)  ? 

9.  How  much  will  it  cost,  at  30  cents  per  sq.  yd.,  to  plaster 
the  bottom  and  sides  of  a  cubical  cistern  that  will  hold  300 
barrels  ? 

10.  A  miller  wishes  to  make  a  wagon-box  large  enough  to 
hold  100  bushels,  having  the  length  3  times  the  width  and 
height.     What  will  be  its  dimensions? 

11.  Which  has  the  greater  surface,  a  cube  whose  solid 
contents  are  13824  cubic  feet,  or  a  rectangular  solid  having 
the  same  solidity,  whose  height  is  half  its  length  and  whose 
width  is  three-fourths  its  length?     How  much? 

12.  If  a  cubic  metre  contains  61026.048  cubic  inches,  what 
is  the  length  of  a  linear  metre  ? 


SBULAR  FIGURES. 

635.  The  truth  of  the  following  principles  can  be  shown 
by  geometry: 

Principles. — 1.  Similar  solids  are  to  eacJi  otiier  as  the  cubes 
of  ilieir  like  dimensioiw.     Hence, 

2.  The  corresponding  dimensions  of  similar  solids  are  to  each 
oOier  as  the  enhe  roots  of  their  volumes. 


CUBE   ROOT.  335 

1.  If  a  globe  4  inches  in  diameter  weighs  8  lb.,  what  will 
be  the  diameter  of  a  similar  one  that  weighs  125  lb.? 

PROCESS.  Analysis. — Since   the   corre- 

1  T  <<'  .'  ^'19^    /'1^         sponding  dimensions  of   similar 

■'■]''   V    "       V  /         Rolids    are    proportional    to    the 

4  :  ^  ;  ;       2  :  5    (2)         cube  roots  of  these  volumes,  we 

X  or  diameter  is  10  in.  ^'^^^  ^^^.  ^»a°ieter  of  the  smaller 

globe  4  inches  :  the  diameter  of 
the  larger  globe  x  :  :  the  cube  root  of  the  weight  of  the  smaller  globe 
f  8  :  the  cube  root  of  the  weight  of  the  other  globe  ^125.  (1).  Ex- 
tracting the  cube  root  of  8  and  125,  and  we  have  Prop.  (2).  Whence 
solving,  the  diameter  is  10  inches. 

2.  If  a  ball  5  fl.  in  diameter  weighs  800  lb.,  what  will 
Ix}  the  diameter  of  a  similar  ball   which  weighs  3  T.  4  cwt.  ? 

3.  If  a  globe  of  gold  1  inch  in  diameter  is  worth  SI 25, 
what  will  be  the  value  of  one  3  inches  in  diameter? 

4.  If  a  cubical  bin  8  ft.  long  will  hold  411.42  bu.,  what 
must  be  the  dimensions  of  a  similar  bin,  that  will  hold  1000 
bushels  ? 

5.  A  ball  3  feet  in  diameter  weighed  2000  lb.  What  will 
be  the  diameter  of  one  that  weighs  1000  lb.? 

6.  Tlie  dimensions  of  a  cubical  bin  were  such  that  it  would 
contain  1000  bushels  of  wheat.  How  would  the  dimensions 
of  a  similar  bin  that  would  hold  8000  bushels  compare  with 
the  dimensions  of  such  a  bin? 

7.  The  diameters  of  two  spheres  are  respectively  4  and 
12  inches.  How  many  times  the  smaller  sphere  is  the 
larger? 

8.  Tliree  women  own  a  ball  of  yarn  4  inches  in  diameter. 
How  much  of  the  diameter  of  the  ball  m^\^t  fM'h  wiii<l  off, 
so  that  they  may  share  equally? 

9.  A  stack  of  hay  in  the  form  of  a  pyruuiiil  12  iX.  high, 
contained  8  tons.  How  high  must  a  similar  stack  be,  that 
it  may  contain  60  tons? 


(CI 


GRESSIONS 


— >^>?C 


536.    1.  How  docs  each  of  the  numbers  2,  4,  6,  8,  10,  12, 

comjwire  with  the  number  that  follows  it? 

2.  How  may  each  of  the  numbers  4,  6,  8,  etc.,  be  obtained 
from  the  one  that  precedes  it? 

3.  How  does  each  of  the  numbers  2,  5,  8,  11,  14,  17,  com- 
pare with  the  number  that  follows  it?  How  with  the  one 
that  precedes  it? 

4.  Write  in  succession  some  numbers  beginning  with  3 
having  a  common  difference  of  2. 

5.  Write  a  series  of  numbers  beginning  at  4,  and  having  a 
common  difference  of  4. 

6.  Write  a  series  of  numbers  beginning  with  25,  and  de- 
creasing by  the  common  difference  4. 

7.  How  does  each  of  the  numbers  2,  4,  8,  16,  32,  etc.,  com- 
pare with  the  one  that  follows  it  ?  How  may  each  be  obtained 
from  the  one  that  precedes  it? 

8.  Write  a  series  of  numbers  beginning  with  2  and  increas- 
ing by  a  common  multiplier  3. 

9.  AVrite  a  series  of  numbers  beginning  with  5,  and  increas- 
ing by  a  common  multiplier  5. 


DEFINITIONS. 

537.  A  Series  of  numbers  is  numbers  in  succession,  each 
derived  from  the  preceding  according  to  some  fixed  laws. 

(38G) 


ARITHMETICAL   PROGRESSION.  387 

538.  The  first  and  last  terms  of  a  series  are  called  the 
extremes,  tlie  intervening  terms  the  means. 

Thus,  in  the  series  2,  4,  C,  8,  10,  the  numbers  2  and  10  arc  the  ex- 
tremes and  the  others  arc  the  means. 

539.  An  Ascending  8et*ies  is  one  in  which  the  num- 
l)ers  increase  regularly  from  the  first  term. 

Thus,  2,  6,  8,  11,  14,  17,  20,  etc.,  is  an  ascen3ing  series. 

540.  A  Uescending  Series  is  one  in  which  the  num- 
bers decrease  regularly  from  the  first  term. 

Thus,  48,  24,  12,  6,  3,  is  a  descending  series. 


ARITHMETICAL  PROGRESSION. 

541.  An  Arithmetical  Progression  is  a  series  of 
nuinl)ers  which  increase  or  decrease  by  a  constant  common 

<lijference. 

Thus,  5,  9,  13, 17,  21,  etc.,  is  an  arithmetical  progression  of  which 
the  common  difference  is  4. 

542.  1.  The  first  term  of  an  arithmetical  series  is  3  and 
the  common  difference  is  2.     What  is  the  7th  term? 

PROCESS.  Analysis. — S  i  n  c  o 

,.„.    rt        /»  -r»  the  common  difference 

(  om.  diff. ,  2  X  6    =  12  ^  j^  2,  the  second  term  in 

I'irst  t«'rm,  3  -f-  12  =  15,  the  7th  term.       equal  to  the  first  ph:s 

once  the  common  differ- 
iici-,  thf  third  term  is  equal  to  the  first  plus  ticice  the  common  differ- 
iiw,  the  fourth  term  is  equal  to  the  first  term  plus  three  times  the  com- 
lon  difference.  Hence,  the  seventh  term  will  be  equal  to  the  first  term 
'us  six  times  the  common  difference,  which  is  15. 

Rule. — Any  iertn  of  an  ariOimetical  progremon  is  equal  to 
(he  first  term,  iticreased  by  the  common  difference  mtUtiplfed  by  a 
number  one  less  tJian  the  number  of  terms. 
22 


338  PROGRESSIONS. 

2.  The  first  term  is  10  and  the  common  difference  5.  What 
is  the  10th  term?     Prove  it. 

3.  The  first  term  is  H  and  the  common  difference  is  8. 
What  is  the  25th  term  ? 

4.  A  boy  agreed  to  work  for  50  days  at  25  cents  the  first 
(lay,  and  an  increase  of  3  cents  per  day.  What  were  his  wages 
the  hist  day? 

».  A  body  falls  16^  feet  the  first  second,  3  times  as  far 
the  second  second,  5  times  as  fiir  the  third  second.  How 
far  will  it  fall  the  seventh  second? 

6.  An  arithmetical  series  has  1000  terms,  the  first  term  of 
which  is  75  and  the  common  differenct^  5.  What  is  the  last 
term? 

7.  Find  the  sum  of  an  arithmetical  scries  of  which  the  first 
term  is  2,  the  common  difference  3,  and  the  number  of  terms  7. 

PROCESS.  Analysis. — By  examining  the  series  2, 

2  4-  r  G  X  3^  -^  2  0        5,  8,  11, 14,  17,  20,  it  is  evident  that  the 

'^  ftA       oo        average  term  is  11,  for  if  half  the  sum 

^  -r  ^^  =^  ^^         of  any  two   terms  equidistant  from  the 

22  -T-    2  =  11         extremes  be  found  it  will  be  11,  and  in 

1  1  X     7  :^=  7  7         general  in  any  arithmetical  progression 

the  average  term  is  equal  half  the  sum 

of  the  extremes  or  any  two  terms  equidistant   from   the  extremes. 

Since   the  first  term  is  2  and  the  common  difference  3,  the  last  terra 

is  found  by  the  previous  rule  to  be  20.    The  sum  of  the  extremes  is 

therefore  22,  which,  divided  by  2,  gives  the  average  term.     And  since 

there  are  7  terms,  the  sum  will  be  7  times  the  average  term,  or  77. 

Rule. — To  find  the  mm  of  an  ariHimdical  series  :  3fidtiply 
half  the  sum  of  the  ejctremes  by  the  number  of  terms. 

8.  What  is  the  sum  of  an  arithmetical  series  composed  of 
50  terms,  of  which  the  first  term  is  2  and  the  common  differ- 
ence 3? 

9.  What  is  the  sum  of  a  series  in  which  the  first  term  is 
1^  the  common  difference,  y^^j,  and  the  number  of  terms  100? 


GEOMETRICAL   PROGRESSION.  339 

10.  A  man  walked  15  miles  the  first  day,  and  increased  his 
rate  3  miles  per  day  for  10  days.  How  fur  did  he  walk  in 
the  eleven  days? 

11.  How  many  strokes  docs  a  clock  strike  in  12  hours? 

12.  A  person  had  a  gift  of  3100  per  year  from  his  birth 
until  he  became  21  years  old.  These  sums  were  deposited 
in  a  bank  and  drew  simple  interest  at  6^.  How  much  was 
due  him  when  he  became  of  age? 


GEOMETRICAL  PROGRESSION. 

543.  A  Geametrical  Progression  is  a  series  of 
numbers  which  increase  or  decrease  by  a  constant  multiplier 
or  ratio. 

Thus,  5,  10,  20,  40,  80,  etc.,  is  a  geometrical  progression,  of  whicli 
the  multiplier  or  ratio  ia  2. 


WRITTEN     EXERCISES, 

544.    1.  The  first  term  of  a  geometrical  series  is  3  and  the 
multiplier  or  ratio  is  2.     What  is  the  5th  term? 


Analysis.— Since   the   multiplier   is   2,  the 

2*=  16  second  term  will  be  3X2,  the  third  3X2X2 

3X16  =  48       or  3  X  2-,  the  fourth  3  X  2^  X  2  or  3  X  2-''  and 

^  the  fifth  3  X  2^  X  2  or  3  X  2*,  that  is,  tlie  fiflh 

term  is  equal  to  the  first  term  multiplied  by  the  ratio  raised  to  the 

fourth  powt  I . 

Rui^. — Auy  trrm  oj  n  (jnmutriral  jrrogression  i«  equal  to  the 
first  term,  multiplied  by  Vic  ratio  raiMcl  to  a  pofwer  one  less  than 
the  number  of  the  term. 

2.  The  first  term  of  a  geometrical  progression  is  10,  the 

ratio  :^.     Wlijit  i>!  tho  (>th  torni? 


340  PROGRESSIONS. 

3.  The  first  term  of  a  geometrical  progression  is  10,  the 
ratio  4,  and  the  number  of  terms  6.     What  is  the  6th  term? 

4.  If  a  farmer  should  hire  a  man  for  10  days,  giving  him 
5  cents  for  the  first  day,  3  times  that  sum  for  the  second 
day,  and  so  on,  what  would  be  his  wages  for  the  last  day? 

6.  If  the  first  term  is  $100  and  the  ratio  1.06,  what  is 
the  6tli  term  ?  Or,  what  is  the  amount  of  8100  at  compound 
interest  for  5  years  at  6%  ? 

6.  What  is  the  amount  of  $520  for  6  years,  at  5^  com- 
pound interest? 

7.  What  is  the  sum  of  a  geometrical  series,  of  which  the 
first  trnii  is  5,  the  ratin  8,  and  the  number  of  terms  5? 

l'HocK^^J^.  ANALYSIS.  —  Sincc     in 

_       o  <        A  n,-      r^     r^»    ^  this  sefies  the  first  term  is 

5  X  8 1  =  4  0  D.  the  5th  term.  5   ,^^  ^^.^  3   ^„^  .^^ 

3X405 5  number  of  terms  5,  their 

3_^ =60  5,  the  sum.  g„ni     ^^y    he    obtaine<i 

by  the  following  process, 
which  illustrates  the  formation  of  the  rule: 

Series  5  +  15  +  45  +  135  +  405 
3  times  Series  15  +  45  +  135  +  405  +  1215 

2  times  Series        =  1215—  5 

Series        =  ^^i^ 

Rule. — The  sum  of  a  geomdncal  series  is  equal  to  Uie  differ- 
ence between  the  first  tenn^  and  the  product  of  the  last  term  hij  the 
raiiOf  divided  by  Hie  difference  between  the  ratio  and  1. 

8.  The  extremes  of  a  geometrical  progression  are  4  and 
1024,  and  the  ratio  4.     What  is  the  sum  of  the  series  ? 

9.  The  extremes  are  \  and  fff  and  the  ratio  2^.  What 
is  the  sum  of  the  series? 

10.  What  is  the  sum  of  the  series  in  which  the  first  term 
is  2,  the  last  term  0,  and  the  ratio  ^;  or  what  is  the  sum  of 
the  infinite  series  2,  1,  |,  \,  -|,  -Jjr,  -^,  etc? 


n 


MENSURATION 


DEFINITIONS. 

545.  Mensuration  treats  of  the  measurement  of  lines, 
surfaces,  and  solids. 

546.  A  Line  is  that  which  has  length  only. 

547.  A  Sti'nUfht  Line  is  a  line  that 
does  not  change  its  direction. 


Straight  Line. 


548.  A  Curved  Line  is  a  line  that 
changes  its  direction  at  every  point. 

549.  Parallel  Lines  are  such  as  are* 
equidistant  throughout  their  whole  extent. 


Curvotl  Lines. 


Parulkl  Lines. 


550.  A  Plane  Surface  is  a  surface  such  that  a  straight 
line  joining  any  two  points  of  it  is  wholly  in  the  surface. 

551.  A  Curved  Surface  is  a  surface  such  that  no  part 
of  it  is  a  plane  surface. 


552.  An  Anf/lf 

two  lines  that  meet. 


the   divorirence   of 


Anglo. 


553.  A  Itif/ht  Anffle  is  the  angle 
formed  when  one  straight  line  meets  another 
making  the  adjacent  angles  equal. 

The  lines  arc  prrjiendicidar  to  each  other 
when  a  right  angle  is  formed. 


Two  Bight  AnglM. 
(Ml) 


342 


MENSURATION. 


Aciito  Aogle. 


Obtuso  Anglo. 


Triangle. 


ZI3 

QnadriUteral. 

ZZ7 

ParaUelosTam. 


Uoctanglc. 


554.  An  Acute  Angle  is  an   angle 

which  is  less  than  a  right  angle. 

555.  An  Obtuse  Angle  is  an  angle 
which  is  greater  than  a  right  angle. 

556.  The  Vertex  of  an  anprle  is  the 
point  where  the  sides  meet. 

557.  A  Triangle  is  a  figure  which  has 
three  sides  and  three  angles. 

558.  A  Quadrilateral  is  a  figure 
bounded  by  four  sides. 

559.  A  Parallelogram  is  a  quadri- 
lateral whose  opposite  sides  are  parallel. 

5(>0.  A  Rectangle  is  a  parallelogram 
whose  angles  are  right  angles. 

561.  A  Polygon  is  a  plane  figure 
bounded  by  straight  lines. 

562.  A  Circle  is  a  plane  figure  bounded 
by  a  curved  line  every  point  of  which  is 
equally  distant  from  a  point  within  called 
the  center. 

563.  The  Circumferetice  is  the  line 
which  bounds  the  circle. 

564.  A  Bad  ins  of  a  circle  is  a  straight 
line  drawn  from  the  center  to  the  circum- 
ference. 

565.  A  Diameter  of  a  circle  is  a 
straight  line  drawn  through  the  center  and 
terminating  at  both  ends  in  the  circumfer- 
ence. 


MENSURATION. 


343 


Buac. 


5(>6.  The  Base  of  a  figure  is  the  side 
on  whieli  it  is  assumed  to  stand. 

567.  The  Altitude  of  a  figure  is  the 
per|x?ndicular  distance  between  the  base  and 
the  highest  point  opposite  it. 

56S.  A  jyiagonal  of  a  figure  is  a 
straight  line  joining  the  vertices  of  two 
angles  not  adjacent. 

560.  The  Perinieter  of  a  figure  is  the  length  of  the 
lines  that  bound  it. 

570.  The  Area  of  a  surface  is  the  definite  amount  of 
surface  it  contains. 


^S^ 


MEASUREIMENT  OF  LINES. 

571.  It  can  be  shown  by  geometry  that  the  circumference 
of  a  circle  is  3.1416  —  times  its  diameter. 

For  ordinary  measurements  it  is  sufficiently  accurate  to  consider 
the  circumference  3f  times  the  diameter. 

Rule. — 1.   Tlie  circumference  is  equal  to  the  diameter  muUi- 
plied  by  S.UIQ. 

2.   The  circumference  divided  by  3. 1416  is  equal  to  the  diameter. 


wnr  T  1 1:  a    i:xEitc  i  SES. 

572,    1.  What   i-   the  oironniforonpo  of  a  oirHo   10  foet   in 
diameter? 

2.  What  is  the  circumference  of  a  circle  45  feet  in  diameter  ? 

3.  How  far  is  it  around  a  circular  lake  that  is  300  rods  in 
diameter  ? 

4.  What  is  the  circumference  of  a  circle  whose  radius  is  20 
rods? 


344  MENSURATION. 

5.  What  is  the  circumference  of  a  circle  whose  radius  is  5 
feet  ()  inches? 

6.  What  is  the  diameter  of  a  circle  whose  circumference 
is  318.5  rods? 

7.  What  is  the  radius  of  a  circle  whose  circumference  ia 
1284  rods? 


MEASURK^fKXT  OF  SURFACES. 

57*i.  To  compute  the  »roa  of  a  parallelo^^rain. 

The  truth  of  the  following  principle  has  been  shown  already: 

Principle. — The  area  of  any  rectangular  figure  is  equal  to 
Hie  product  of  its  length  by  ite  breadtJi  or  altitude. 

0  By  examining  the  figure  A,  B,  C,  D,  it  will 

be  seen  that  it  ia  equal  to  E,  F,  D,  C,  and  that 

any  oblique  parallelogram  is  equal  to  a  rectangu- 

_/-J        lar  parallelogram  of  the  same  base  and  altitude. 

Therefore, 


i 


Rule. — The  area  of  any  parallelogram  is  equal  to  the  prod- 
uct of  the  base  multiplied  by  the  altitude. 


WRITTEN    EXEltCISBa, 

674.  1.  How  many  .square  feet  are  there  in  a  parallelo- 
gram, whose  length  is  40  feet  and  altitude  13  feet? 

2.  What  is  the  area  of  a  parallelogram  whose  base  meas- 
ures 7  feet  and  whose  altitude  is  3  feet  8  inches? 

3.  What  is  the  area  of  a  field  in  the  form  of  a  parallelo- 
gram, whose  length  is  30  rods  and  the  perpendicular  distance 
between  the  sides  is  24  rods? 

4.  What  is  the  area  of  a  pamllelogram  whose  length  is 
35  feet  and  whose  altitude  is  15  feet? 


MKN.^UliATION.  345 


675.  To  compute  the  area  of  a  triangle. 

If  C  E  be  drawn  parallel  to  the  base  of  the  ^ .  ] 

triangle,  and  B  E  be  drawn  parallel  to  A  C, 

the   parallelogram   ABEC  will  be  formed, 

of  which  the  original  triangle  is  one-half.     In 

the  same  manner  it  can  be  shown  that  every 

triangle  is  one-half  of  a  parallelogram  of  the  same  base  and  altitude. 

Therefore, 

Rule. — The  curea  of  a  triangle  is  equal  to  one-half  the  prod- 
uct of  the  base  by  the  altitude. 

When  the  three  sides  are  given,  the  following  is  the  rule: 

Rule. — From  half  Hie  stim  of  the  three  sides  subtract  each  side 
separately.  Midtiply  together  the  half  sum  and  the  three  remain- 
derSy  and  extract  Vie  square  root  of  Uie  product.  The  resvU  mil 
be  the  area  of  the  triangle. 

WRITTEN    EXERCISES, 

576.  1.  What  is  the  area  of  a  triangle  whose  base  is  24 
feet,  and  whose  altitude  is  18  feet? 

2.  What  is  the  area  of  a  triangle  whose  base  is  21  feet 
and  whose  altitude  is  12  feet? 

3.  What  is  the  cost,  at  $850  per  acre,  of  a  triangular  piece 
of  ground,  the  three  sides  of  which  are  in  a  ratio  of  5,  6 
and  8,  and  whose  shortest  side  is  120  feet? 

4..  What  is  the  area  of  a  triangle,  the  three  sides  of  which 
are  respectively  180  feet,  150  feet,  and  80  feet? 

5.  A  liouse  is  32  feet  wide,  and  the  rafters  are  20  feet 
long  on  each  side,  exclusive  of  any  projections.  What  will 
the  lumber  cost  at  $22.50  per  M,  whicli  will  inclose  both 
gable  ends  of  the  house? 

fi.  What  is  the  area  of  a  triangle  whose  base  is  300  feet, 
and  whose  altitude  is  100  feet? 


346 


MENSURATION. 


To  eompate  the  area  of  n  polj^oii. 


Since  any  figure  may  be  divided  into  tri- 
angles, its  area  will  be  the  area  of  the  triangles 
which  compose  it.     Therefore, 

Rules. — I.  Ttie  area  of  a  trapezium  is 
equal  to  the  dia^/onal,  imdtijiUed  hy  half  the 
sum  of  tJie  perpendiculars  drauii  from  Hie 
vertices  of  tiie  opposite  angl^  to  the  diagonal, 

II.  The  area  of  a  trapezoid  is  equal  to 
the  sum, of  the  parallel  sides  multiplied  hy 
Judf  tJie  aliOude. 

III.  The  area  of  a  regxdar  jx)lygon  is 
equal  to  Vie  perimeter  of  tiie  polygon  mul- 
tiplied by  one -half  the  perpendicular  dis- 
tance from  the  center  to  one  of  tlie  sides  of 
the  jiolygon. 


JicKuIar  Polypcoa. 


WRITTEN    EXERCISES. 

578.  1.  What  is  the  area  of  a  trapezium,  the  diagonal  of 
Avhich  is  110  feet,  and  the  perpendiculars  to  the  diagonal  are 
40  feet  and  60  feet  respectively? 

2.  The  parallel  sides  of  a  trapezoid  are  respectively,  10 
rods  and  8  rods,  and  the  altitude  6  rods.     What  is  its  area? 

3.  A  figure  is  composed  of  8  triangles,  whose  bases  arc 
each  12  ft.  and  their  altitude  12  ft.     What  is  the  area? 

4.  What  is  the  cost,  at  $125  per  acre,  of  a  piece  of  ground 
in  the  form  of  a  trapezoid,  whose  parallel  sides  are  respect- 
ively, 40  rods  and  30  rods,  and  whose  altitude  is  20  rods? 

5.  I  paid  $110  per  acre  for  a  piece  of  ground  in  the  form 
of  a  trapezium.  A  diagonal  line  crossing  it  was  120  rods 
long,  and  the  perpendiculars  drawn  to  the  diagonal  were, 
respectively,  30  rods  and  20  rod?.     What  did  it  cost  me? 


MENSURATION.  347 

570.  To  compote  the  area  of  a  cir.'lo. 

From  the  accompanying  figure  it  i- 
.  vident  that  a  circle  may  be  regarde<l 
as  composed  of  a  large  number  of  tri- 
angles, the  sum  of  whose  bases  forms 
the  circumference  of  the  circle,  and 
whoso  altitude  is  the  radius  of  the 
circle.    Hence, 

Rule. — 1.   TJie  area  of  a  circle  is  equal  to  tlie  circumference, 
multiplied  by  ludf  the  radius;  or, 
2. — Tlie  square  of  tJie  diameter  midtiplied  by  .7854. 


WRITTEN    JCXERClSi:s. 

5M),     1.  What   is  the  area  <if  a  circle  whose  diameter  is 
5  feet? 

2.  What  is  the  area  of  a  circle  whose  diameter  is  8  feet? 

3.  What   is   the  area  of  a  circle  whose  circumference  is 
120  rods? 

4.  What  is  the  area  of  a  circle  whose  circumference  is  100 
feet? 

5.  A  gentleman  discovered  that  the  distance  around  a  cir- 
cular pond  was  320  rods.     AVhat  was  its  area? 

6.  If  a  horse  is  tethered  to  a  stake  by  a  rope  15  rods  long, 
over  how  much  surface  can  he  graze? 

7.  How  long  must  a  ro|XJ  be  that  a  horse  o'ln  irraze  on 
just  an  acre? 

8.  Tlie  area  of  a  circle  is  113.0976  square  rods.     What  is 
its  diameter? 

9.  The  round-house  of  the  P.  and  S.  Railroad  is  350  feet 
in  diameter.     How  much  land  does  it  cover? 

10.  W^hat  is  the  area  of  a  railroad  turn-table  35  feet  in 
diameter? 


348 


MENsri:  \  I  i"N, 


MEASUREMENT  OF  SOLIDS. 


DEFINITIONS. 

681.  A  Solid  or  Body  is  that  which  has  length,  breadth 
and  thickness. 

The  planes  which  bound  a  solid  are  called  its /aces,  and  their  inter- 
sections its  edges. 


^^^ 


Prism. 


Paral  Iclopj  pcdoii. 


Cylinder. 


682.  A  J^ristn  is  a  solid,  having  its  two  ends  equal  poly- 
gons, parallel  to  each  other,  and  its  sides  parallelograms. 

Prisms  are  named  from  the  form  of  their  bases  triangular, 
quadrangular y  pentagonal,  etc 

583.  A  I^arallelopipedon  is  a  solid  whose  opposite 
faces  are  equal  and  parallel  parallelograms. 

584.  A  Cylinder  is  a  regular  solid  bounded  by  a 
uniformly  curved  surface,  and  having  for  its  ends  two  equal 
circles,  parallel  to  each  other. 

The  face  of  any  section  of  a  cylinder  parallel  to  the  base  is  a  circle 
equal  to  the  base. 

585.  A  I^yramid  is  a  solid  whose  base  is  a  polygon 
and  whose  faces  are  triangles,  meeting  at  a  point  called  the 
vertex  of  the  nvi-amid. 


MEASl  ki<.Mi,N T    OF   SOLIDS.  349 

586.  A  Cone  is  a  solid,  whose  l)ase  is  a  circle  and  whose 
Burface  tapers  uniformly  to  a  point  called  the  vertex. 


CoDC.  Fruitum  of  Cone. 

587.  A  Ft*ustuni  of  a  pyramid  or  cone  is  the  portion 
remaining,  after  the  top  has  been  cut  off  by  a  plane  parallel 
to  the  base. 

588.  A  Sphere  is  a  solid,  every  point  of  whose  surface 
is  equally  distant  from  a  point  within,  called 
the  center. 

589.  The  I>ianieter  of  a  sphere  is  a 
straight  line  passing  through  the  center,  and 
terminating  in  the  surface  at  both  ends. 

59().  The  Hadius  of  a  sphere  is  one- half  the  diameter, 
or  the  distance  from  the  center  to  the  surface. 

591.  The  Circumfevence  of  a  sphere  is  the  greatest 
distance  around  the  sphere. 

51>2.  The  Altifufle  of  a  solid  is  the  peri)endicular  dis- 
tance from  its  ])i"h(>st  }X)int  to  the  plane  of  the  base. 


CONVEX  SURFACE  OF  SOLIDS. 

r»93.  The  Convex  Surface  of  a  solid,  is  all  its  surface 
except  it^  base  or  bases.     The  entire  convex  surface  includes 

tho  Jirt'ti  of  the  luisrs  iils<». 


>0  MENSIK  A'lloN'. 

59-4.  To  find  the  convc>i^  Nurfac^*  oi'a  pri^in  or  cylinder. 

It  Ih  evident  that  if  :i  priKin  or  cylinder  were  1  inch 
liitrh,  its  convex  surface  would  contain  as  many  square 
u\u\<  of  surface  as  there  were  units  in  the  perimeter  of 
tlu-  hase;  and  if  it  were  2  inches,  3  inches,  or  4  inches 
high,  the  convex  sjirface  would  contain  2,  3,  or  4  times 
t^o  nnnilxT  of  units  in  the  jx'rinieter  of  the  base. 
II:..    11:     following 

Rule. — Multiply  Uie  perimeter  of  the  base  by  the  altitude. 


WRITTBy    BXBnciSES. 


vex  surface  of  a  cylinder  whose 


length  5  feet? 


595.    1.  Wiat  is  til 
diameter  is  2  feet  and 

2.  What  is  the  convex  surface  of  a  quadrangular  prism 
whose  sides  are  each  2h  feet  and  wliose  height  is  i  feet  ? 

8.  AVhat  is  the  convex  surface  of  a  triangular  prism  whose 
siiU  -  art'  tiuh  6  feet  and  whose  altitude  is  8  feet? 

4.  AVluit  is  the  entire  surface  of  a  cylinder  which  is  5  feet 
in  length,  and  whose  base  is  2  feet  in  diameter? 

5.  What  is  the  convex  surface  of  a  piece  of  timber  in  the 
form  of  a  triangular  prism,  which  is  18  feet  long  and  the  sides 
of  whose  base  are  10  inches,  14  inches,  and  18  inches? 


596.  To  find  the  conTex  sarfacc  of  a  pyramid  or  cone. 

It  is  evident  that  the  convex  surface  of  any  pyra- 
mid is  composed  of  triangles,  and  the  convex  surface 
of  a  cone  may  also  be  assumed  to  be  made  up  of  an 
infinite  number  of  triangles.  The  bases  of  these 
triangles  form  the  perimeter  of  the  solid,  and  their 
height  is  the  slant  height  of  the  solid.  Therefore 
the  following  is  the  rule: 

Rule. — Multiply  the  perimeter  of  the  base  by  one-half  the  dant 
height. 


MFASTTRKMENT   OF   SOLIDS.  351 


WRITTEN     EXERCISES, 


507.  1.  What  is  the  convex  surface  of  a  quadrangular 
pyramid,  whose  base  is  15  feet  square  and  the  slant  height  18 
fJet? 

2.  What  is  the  convex  surface  of  a  cone  whose  diameter  at 
the  base  is  12  feet  and  whose  slant  height  is  20  feet? 

3.  What  is  the  convex  surface  of  a  cone  whose  base  is  20 
feet  in  diameter  and  whose  slant  height  is  20  feet? 

4.  What  is  the  cost  of  painting  a  church  steeple,  the  base 
of  which  is  an  octagon  6  feet  on  each  side,  and  whose  slant 
height  is  80  feet,  at  8.30  j^er  square  yard? 

5.  How  many  feet  of  convex  surface  are  there  on  a  cone,  the 
base  diameter  of  which  is  6  feet  and  whose  slant  height  is  9^ 
feet? 

6.  How  many  feet  of  convex  surface  are  there  on  a  pyra- 
mid whose  base  is  10  feet  square  and  whose  slant  height  is 
20  feet? 

7.  How  many  feet  of  convex  surface  are  there  on  a  cone 
whose  base  is  8  feet  in  diameter  and  whose  slant  height  is 
f)  feet? 

8.  What  is  the  convex  surface  of  a  cone  whose  base  is  10 
feet  in  diameter  and  whose  slant  height  is  10  feet? 

59S.  To  find  the  convex  sarfkce  of  a  fVnNtnni  of  a 
p>  ruuiid  or  «one. 

1 1  i-  » vident  that  the  convex  surface  of  a  frustum 
of  a  pyramid  is  comix)fted  of  trapezoids,  the  sum  of 
whose  parallel  sides  forms  the  perimeter  of  the  bases, 
and  whose  altitude  is  the  slant  height  of  the  frustum; 
and  the  convex  surface  of  a  cone  may  be  assumed  to  be  made  of  an 
infinite  number  of  trapezoids.     Hence, 

RuLi-:. — ^fuU^ply  halfUie  sum  of  the  perimeter  of  the  two  bases 
by  the  slant  height. 


MFN<ri: A  rioN 


II  /;  /  /  /  /;  V     /;  a  /;  /.■  <  ;  >  y.  s 


599.  1 .  How  many  feet  of  convex  surface  are  there  in  the 
frustum  of  a  cone  whose  slant  height  is  8  feet,  the  diameter 
of  whose  lower  base  is  12  feet  and  up])er  base  8  feet? 

2.  Wliat  is  the  convex  surface  of  the  frustum  of  a  pyramid 
the  shint  height  of  which  is  25  feet,  whose  lower  base  is  40  feet 
square,  and  whose  upper  base  is  20  feet  square  ? 

3.  What  did  it  cost,  at  $.15  per  sq.  yd.,  to  paint  the  con- 
vex surface  of  a  vat  which  was  10  feet  in  diameter  at  the 
bottom  and  8  feet  at  the  top,  the  slant  height  of  which  was 
12  feet? 

4.  What  is  the  convex  surface  of  a  vat,  the  base  of  which 
is  0  feet  square  whose  top  is  8  feet  square  and  whose  slant 
height  is  10  feet? 

5.  What  would  the  lumber  cost  at  840  per  M,  to  build 
such  a  vat  if  the  sides  were  of  IJ  inch  plank,  and  the  bottom 
was  2  inch  plank  ? 

GOO.  To  find  (he  con%ex  surOu^  or  a  sphere. 

Tlie  convex  surface  of  a  sphere  is  computed,  according  to  geomet- 
rical principles,  by  the  following  rule: 

Rule. — 1.  Multiply  the  diameter  by  the  circumference. 
2.  Multiply  the  square  of  the  diameter  by  3.1416. 

EXEBCI8JE8. 

601.  1.  What  is  the  convex  surface  of  a  sphere  whose 
diameter  is  15  inches? 

2.  What  is  the  convex  surface  of  a  spherical  cannon-ball  8 
inches  in  diameter? 

3.  What  is  the  convex  surface  of  a  base-ball  whose  circum- 
ference is  9^  inches? 


MEASUREMENT  OF  SOLIDS.  363 

4.  What  is  the  convex  surface  of  a  sphere  whose  circum- 
ference is  12  feet? 


VOLUME  OF  SOLIDS. 

602.  The  Voltnne  of  any  body  is  the  number  of  solid 
units  it  contains. 


^ii 


003.  To  find  the  voluiue  oi*  a  prism  or  cyliiKler. 

It  is  evident  that  if  a  prism  or  cylinder  were  1  inch 
high,  it  would  contain  as  many  cubic  inches  as  there 
were  square  inches  in  the  area  of  the  base;  and  if  it 
were  2  inches,  3  inches,  or  4  inches  high,  the  vohime 
would  be  2  or  3  or  4  times  as  much.  Hence  the  fol- 
lowing is  the  rule: 

Rule. — Multiply  the  area  of  the  base  by  the 
altitude. 

WJ{ITTJ^:y     EXERCISES. 

604.  1.  What  are  the  solid  contents  of  a  prism  whose 
base  is  12  inches  square  and  whose  height  is  2  feet  ? 

2.  What  is  the  volume  of  a  cylinder  whose  diameter  is  IJ 
feet  and  whose  length  is  4  feet? 

3.  What  would  be  the  cost  of  a  piece  of  timber  20  feet  long, 
18  inches  wide  and  12  inches  thick  at  $.30  per  cubic  foot? 

4.  What  will  be  the  eajxicity  in  bushels  of  a  square  bin  the 
base  of  which  was  8  feet  square,  and  the  height  of  which  was 
9  feet  on  the  inside  ? 

5.  How  many  gallons  of  water  will  a  vat  in  the  form  of  a 
rylinder  hold,  whose  inside  dimensions  are — base  8  feet  in 
diameter,  height  7  feet? 

G.  How  much   would   the  wheat  be  worth  at  81.85  jMjr 

bushel,  which  would  just  fill  a  bin  the  base  of  wVwh  '-'  1 '» 

feet  square,  andy the  height  of  which  is  12  feet? 

23 


0'J4  MENSrUATIoX. 

005.  To  And  the  Yoliiiiie  of  a  pyramid  or  cone. 

It  can  be  shown  by  geomelry  that  a  pyramid  or  cone  is  one-third 
of  a  prism  or  cylinder  of  the  same  base  and  altitude.  Hence  the  fol- 
lowing is  the 

KuLE. — Multiply  die  area  of  the  base  by  one-Unnl  i>j  Ihc  altitude. 


WRITTEN    EXERl   i  ^  1^. 

600.  1.  What  are  the  solid  contents  of  u  cone,  th(i  diam- 
eter of  whose  base  is  6  feet  and  whose  altitude  is  9  feet  ? 

2.  What  are  tlie  solid  contents  of  a  pyramid  whose  base  is 
80  feet  square  and  whose  altitude  is  60  feet? 

8.  If  a  -cubic  foot  of  granite  weighs  165  lb.,  what  is  the 
weight  of  a  granite  cone  the  diameter  of  whose  base  is  6  feet 
and  whose  altitude  is  8  feet  ? 

4.  What  is  the  weight  of  a  marble  pyramid  whose  base  is 
4  feet  S(juare  and  whose  altitude  is  8  feet,  if  a  cubic  foot 
of  marble  \Yeighs  171  pounds  ? 

007.  To  find  the  Tolame  of  a  fk'nstom  or  a  pyramid 
or  coue. 

It  can  be  shown  by  geometry  that  the  frustum  of  a  pyramid  or 
cone  is  equal  to  three  pyramids  or  cones,  having  for  their  bases,  re- 
spectively, the  upper  base  of  the  frustum,  its  lower  base,  and  a  mean 
l)roportional  between  the  two  bases.     Hence  the  following  is  the 

Rule. — To  the  sum  of  Hie  areas  oftlie  two  ends  add  the  square 
root  of  Hie  product  of  tliese  areas,  and  multiply  Hie  residt  by  one- 
third  of  the  altitude. 

EXERCISES. 

608.  1.  What  is  the  volume  of  a  frustum  of  a  pyramid 
the  lower  base  of  which  is  20  feet  square,  the  upper  base  10 
feet  square  and  the  altitude  20  feet? 


i 


MEASUREMENT   OF  SOLIDS.  355 

2.  What  are  the  solid  contents  of  the  frustum  of  a  cone 
wliose  upjHjr  base  is  5  feet  in  diameter,  whose  lower  base  is 
H  feet  in  diameter,  and  whose  altitude  is  7  feet? 

3.  A  tree  was  3  feet  in  diameter  at  the  butt  and  its  diam- 
eter at  a  height  of  40  feet  was  1  foot.  What  were  the 
cubical  contents  of  that  portion  of  the  tree  ? 

4.  A  vat  whose  inside  measurements  were  as  follows — 
diameter  of  the  bottom  12  feet,  diameter  of  the  top  10  feet, 
height  9  feet— was  filled  with  water.  How  many  gallons 
did  it  contain? 

GOO.  To  fiud  the  Tolanie  or  contents  or  a  sphere. 

A  sphere  may  be  regarded  as  composed  of  pyra- 

mills  whose  bases  form  the  surface  of  the  sphere,  ^^r^^^ 

and  whose  altitude  is  the  radius  of  the  sphere.  ^^^,       1^ 

Hence  the  following  is  the  ^Bm-'-""'''^^M 

Rule.— 1.  MuUlply  the  convex  mrface  by  ^M^|j|E 
one -third  of  the  radius;  or,  ^^^^^^ 

2.  Multiply  ilie  cube  of  Hie  diameter  by  .5236. 


EXERCISES. 


low  mnnv 


610.    1.  The  diameter  of  a  sphere  is  5  foot.     Hr 
cul)ic  feet  does  it  contain? 

2.  Find  the  contents  of  a  sphere  whose  diameter  is  8  feet. 

3.  The  circumference  of  a  sphere  is  9.4248.  AVhat  are  its 
cubiail  contents? 

4.  A  cubic  foot  of  ciist-iron  weighs  about  450  pound:?. 
Wliat  is  the  weight  of  a  ainnon-ball  whose  diameter  is  18 
inclu's? 

5.  Wliat  are  the  cubical  content?^  of  n  spherical  vessel  the 
diameter  of  which  is  2.J^  feet? 

().  How  many  cubic  feet  are  there  in  a  spherical  body  whose 
diameter  is  25  feet? 


356  MISCELLANEOUS   EXAMPLES. 


MISCELL.^J^EOUS  EXAMPLES. 

611.  1.  If  5  men  can  do  a  piece  of  work  in  12  days,  how 
long  will  it  take  6  men  to  do  the  same  work? 

2.  If  5  barrels  of  apples  cost  $7.50,  what  will  8  barrels 
cost  at  the  same  rate? 

3.  It  required  20  men  to  load  a  vosscl  in  G  days,  how 
many  men  would  it  require  to  load  it  in  1^  days? 

4.  A  steamboat  sailed  42^  miles  in  2J  hours.  How  far 
did  she  sail  in  20  minutes? 

5.  If  6  men  can  dig  28  rods  of  ditch  in  1  day,  how  many 
men  will  it  require  to  dig  56  rods  in  }  of  a  day? 

6.  If  I  of  a  yar<l  of  broadcloth  cost  83f ,  what  will  ^  of  n. 
yard  cost? 

7.  If  it  costci  $50  to  sup|X)rt  a  family  of  8  persons  for  2^ 
weeks,  what  will  it  cost  to  support  10  persons  3  weeks? 

8.  If  3  pounds  of  tea  are  worth  14  pounds  of  coffee,  and 
5  pounds  of  coffee  are  worth  18  pounds  of  sugar,  and  21 
pounds  of  sugar  are  worth  GO  i)()unds  of  flour,  how  many 
pounds  of  flour  are  equal  in  value  to  7  pounds  of  tea? 

9.  A  farmer  sold  12  firkins  of  butter,  each  containing  56 
pounds,  for  23  cents  a  pound,  and  received  in  payment  5 
pounds  tea  at  85  cents  per  pound,  60  pounds  sugar  at  13 
cents  per  pound,  15  yards  cloth  at  SI.  12^  per  yard,  and  the 
balance  in  money.     How  much  money  did  he  receive? 

10.  A  regiment  of  soldiers  consisting  of  1100  men,  was 
furnished  with  bread  suflScient  to  last  it  8  weeks,  allowing 
each  man  15  oz.  per  day.  If  ^  of  it  was  found  to  be  unfit 
for  use,  how  many  ounces  per  day  shall  each  man  receive  so 
that  the  balance  may  last  8  weeks? 

11.  A  man  being  asked  how  many  sheep  he  had,  replied, 
"  If  I  had  3  times  as  many  as  I  have  and  5  sheep,  I  would 
have  185."     How  many  had  he? 


MISCELLANEOUS   EXAMPLES.  357 

12.  A  man  paid  ^  of  his  money  on  a  debt,  .}  of  the  remain- 
der for  a  suit  of  clothes,  ^  of  the  remainder  for  provisions, 
and  lost  ^  of  the  remainder,  when  he  had  $5  left.  How 
much  had  he  at  first? 

13.  Three  men  engage  to  reap  a  field  of  wheat.  A  can 
do  it  in  15  days,  B  in  18  days  and  C  in  20  days.  In  what 
time  can  they  do  it  together? 

14.  A  farmer  was  offered  31.45  per  bu.  for  his  wheat,  but 
determined  to  have  it  ground  and  sell  the  flour.  It  cost  to 
take  it  to  the  mill  2}  cents  per  bu. ;  the  miller  took  \  for 
grinding;  it  took  4|^  bu.  to  make  a  barrel  of  flour;  he  paid 
45  cents  apiece  for  barrels,  and  it  cost  25  cents  per  barrel 
commission  to  sell  it.  75  bbl.  sold  for  8550  and  25  bbl.  for 
8165.  If  the  refuse  was  sold  for  8100,  did  he  make  or  lose, 
and  how  much  per  hundred  barrels? 

15.  A  farmer  being  asked  how  many  apple-trees  he  had, 
replied,  **  If  I  had  3  times  as  many  and  5  trees  more,  I  would 
have  1358."     How  many  had  he? 

16.  ^  of  A's  money  is  equal  to  J  of  B's,  and  the  difference 
is  88.     How  much  has  each  ? 

17.  A,  B  and  C  hire  a  pasture  for  8170.  A  puts  in  70 
sheep /or  Q\  months,  B  24  cattle  for  4^  months,  C  10  cattle 
and  35  sheep  for  5J  months.  If  2  cattle  oat  as  much  as  7 
sheep,  how  much  should  each  payV 

18.  If  a  pole  10  feet  long,  casts  a  shadow  lo  feet  long, 
what  is  the  length  of  a  pole  which  will  cast  a  shadow  62J 
feet  long  at  the  same  time? 

19.  A's  weight  is  }  that  of  B,  and  C's  welgiit  is  as  much 
as  A*8  and  B's  U>gether.  The  sum  of  their  weights  is  490 
pounds.     What  is  the  weight  of  each? 

20.  J  of  A's  money  is  equal  to  ^  of  B's,  and  the  difToronce 
is  85.     How  much  money  has  each  ': 

21.  The  ages  of  A,  B  and  C,  are  to  each  other  :i-  J.  4 
and  5,  and  their  sum  is  136  years.     What  is  the  age  *>!  tai  ii? 


358  MISCELLANEOUS    EXAMi'i.KS. 

22.  A  boy  bought  a  certain  number  of  apples  at  the  rate 
of  4  for  5  cents,  and  sold  them  at  the  rate  of  3  for  4  cents. 
He  gained  60  cents.     How  many  did  he  buy  ? 

23.  A,  B  and  C  agree  to  build  a  house.  A  and  B  can  do 
the  work  in  32  days,  B  and  C  in  28  days,  and  A  and  C  in 
26  days.  How  long  will  it  take  them  to  do  it  by  working 
together?     How  long  would  it  take  each  to  do  it  alone? 

24.  A  can  build  a  wall  in  10  days,  by  working  12  hours 
a  day,  B  can  build  it  in  9  days,  by  working  10  hours  a  day. 
In  how  many  days  can  both  build  it,  by  working  8  hours  a 
day? 

25.  A  pair  of  horses  is  sold  for  8390.  One  of  them  is 
worth  f  as  much  as  the  other.  What  is  the  value  of 
each? 

26.  A  hind  wheel  of  a  carriage  4  feet  6  inches  high,  re- 
volved 720  times  in  going  a  certain  journey.  How  many 
revolutions  did  the  fore  wheel  make,  which  was  4  feet  high  ? 

27.  The  shadow  of  a  pole  6  feet  long  is  9  inches,  and  the 
shadow  of  a  steeple  at  the  same  time  is  9  feet  long.  What 
is  the  height  of  the  steeple? 

28.  What  is  the  bank  discount  on  a  note  for  $245.30,  due 
in  90  days,  if  discounted  at  6^  ?  ^ 

29.  If  a  man  takes  2  steps  of  30  inches  each  in  3  seconds, 
how  Ibng  will  it  take  him  to  walk  10  miles? 

30.  It  cost  $150  to  support  4  grown  persons  and  3  children 
8  weeks.  What  will  it  cost  to  support  3  grown  persons  and 
8  children  for  the  same  time,  if  3  children  cost  as  much  as 
2  grown  persons? 

31.  A  man  bought  20  bushels  of  wheat  and  15  bushels  of 
corn  for  836,  and  15  bushels  of  wheat  and  25  bushels  of  corn 
for  §32.50?     What  did  he  pay  per  bu.  for  each? 

32.  A  fox  has  120  rods  the  start  of  a  hound.  If  the  hound 
runs  30  rods  while  the  fox  runs  26,  how  fiir  will  the  hound 
run  before  he  overtakes  the  fox? 


mi8cj:llaneol'8  i:xami*i.i:.s.  359 

33.  A  storts  on  a  journey  at  the  rate  of  3  miles  an  hour. 
G  hours  afterward,  B  stiirts  aiter  him  at  the  rate  of  4  miles 
an  hour.     How  far  will  B  travel  before  he  overtakes  A? 

34.  One-fourth  of  a  certain  niunhor  is  10  more  than  ^  of 
it.     AVhat  is  the  number? 

35.  If  to  a  certain  number  you  add  J-  of  itself  and  |  of 
itself,  the  sum  will  Ix)  105.     What  is  the  number? 

36.  If  to  a  certain  number  you  add  15  more  than  |  of 
itself,  the  sum  will  be  40.     What  is  the  number? 

37.  How  many  days  will  it  take  30  men  to  do  a  piece  of 
work,  which  20  men  can  do  in  45  days? 

38.  If  a  man  can  earn  |^  of  a  dollar  in  f  of  a  day,  how 
much  can  he  earn  in  f  of  a  day? 

39.  How  many  yards  of  silk  f  yard  wide,  will  it  take  to 
line  4\  yards  of  broadcloth  1|  yards  wide? 

40.  If  14  ounces  of  wool  make  2\  yards  of  cloth  1  yard 
wide,  how  much  will  it  take  to  make  6J^  yards  IJ^  yards 
wide? 

41.  How  many  tiles  14  inches  long,  will  it  take  to  make 
a  drain  which  is  |^  of  a  mile  long? 

42.  If  3300  placed  at  interest  yields  an  income  of  $18  in 
9  mo^^ths,  how  much  must  be  placed  at  interest  at  the  same 
mte  to  yield  an  income  of  3115  in  6  months? 

43.  If  to  a  certain  number  you  add  \  of  itself,  the  result  will 
be  20  less  than  double  the  number.     What  is  the  numlx'r? 

44.  At  what  time  between  5  and  6  o'clock  will  the  hour 
and  minute  hands  of  a  clock  be  exactly  together? 

45.  Two  soldiers  start  together  for  a  certiiin  fort.  One, 
who  travels  12  miles  per  day,  after  traveling  9  days,  turns 
back  a.s  far  as  the  other  had  traveled  during  those  9  days. 
He  then  turns  and  pursues  his  way  toward  the  fort,  where 
both  arrive  together  18  days  from  the  time  they  set  out  At 
what  mte  did  the  other  travel? 

46.  A  man  agreed  to  execute  a  piece  of  work  in  60  days, 


360  MISCELLANEOUS   EXAMPLES. 

and  employed  30  men  to  perform  the  labor.  At  the  end  of 
40  days  it  was  only  half  finished.  How  many  additional 
laborers  was  he  obliged  to  employ  to  perform  the  work  within 
the  time  agreed  ui)on? 

47.  A  person,  being  asked  the  time  of  day,  replied  that  it 
was  past  noon,  and  that  J  of  the  time  past  noon  was  equal  to 
f  of  the  time  to  midnight.     What  was  the  time? 

48.  A  gentleman  wishes  his  son  to  have  83000  when  he 
is  21  years  of  age.  What  sum  must  be  deposited  at  the  son^s 
birth,  in  a  savings  bank,  which  pays  compound  interest  at 
the  annual  rate  of  6^,  so  that  the  deposit  shall  amount  to 
that  sum  when  the  boy  becomes  of  age? 

49.  A  note  for  $100  was  due  on  Sept.  1st,  but  on  Aug.  11th 
the  maker'  proposed  to  pay  as  much  in  advance  as  will  allow 
him  2  mo.  after  Sept.  1st  to  pay  the  balance.  How  much 
must  be  paid  Aug.  11th,  money  l>eing  worth  6^  ? 

50.  What  sum  must  a  person  save  annually,  commencing 
at  21  years  of  age,  so  that  he  may  be  worth  §25000  when  he 
is  40  years  old,  if  he  gets  6^  compound  interest  on  his 
money? 

51.  If  a  merchant  sells  }  of  an  article  for  what  I  of  it  cost, 
what  is  his  jxnin  j^er  cent.? 

52.  If  goods  are  sold  so  that  f  of  the  cost  is  received  for 
half  the  quantity  of  goods,  what  is  the  gain  per  cent.  ? 

53.  A  man  sold  a  horse  and  carriage  for  $597,  gaining  by 
the  sale  25^  on  the  cost  of  the  horse  and  10^  on  the  cost  of 
the  carriage.  If  f  of  the  cost  of  the  horse  equaled  J  of  the 
cost  of  the  carriage,  what  was  the  cost  of  each  ? 

54.  If  300  cats  can  kill  300  rats  in  300  minutes,  how  many 
cats  can  kill  100  rats  in  100  minutes? 

55.  A  party  of  8  hired  a  coach.  If  there  had  been  4  more 
the  expense  would  have  been  reduced  $1  for  each  person. 
How  much  was  paid  for  the  coach? 

56.  I  sold  goods  at  a  gain  of  20%;.     If  they  had  cost  me 


MlSCELLANl::OU«    EXAMPLES.  361 


S250  more  than  they  did,  I  would  have  lost  20%  by  the 
How  much  did  the  goods  cost  me? 

57.  A  laborer  agreed  to  work  for  $1.25  per  day  and  his 
board,  paying  $  .50  per  day  for  his  board  when  he  was  idle. 
At  the  end  of  25  days  he  received  819.  How  many  days 
was  he  idle? 

58.  A  is  20  years  of  age;  B's  age  is  equal  to  A's  and  half 
of  C's;  and  C's  is  equal  to  A's  and  B's  together.  What  is 
the  age  of  each? 

59.  A  and  B  were  partners  in  a  profitable  enterprise.  A 
put  in  $4500  capital  and  received  f  of  the  profits.  Wliat 
was  B's  capital? 

60.  A  man  spent  $4  more  than  half  his  money  traveling, 
one-half  what  he  had  left  and  $2  more  for  a  coat,  $6  more 
than  half  the  remainder  for  other  clothing,  and  had  $2  left. 
How  much  money  had  he  at  first? 

61.  A  boy  bought  at  one  time  5  apples  and  6  pears  for  28 
cents,  and  at  another  time  6  apples  and  3  pears  for  21  cents. 
What  was  the  cost  of  each  kind  of  fruit  ? 

62.  A  and  B  can  do  a  piece  of  work  in  20  days.  If  A 
does  }  as  much  as  B,  in  how  many  days  can  each  do  it  ? 

63.  A  man  bought  a  farm  for  $5000,  agreeing  to  pay  prin- 
cipal and  interest  in  5  equal  annual  installments.  What  will 
be  the  annual  payment,  including  interest  at  6%? 

64.  A  carriage  maker  sold  two  carriages  for  $300  each. 
On  one  he  gained  25^ ;  on  the  other  he  lost  25^ .  Did  he 
gain  or  lose  by  the  sale?  How  much,  and  how  much  per 
cent.  ? 

65.  If  a  ladder  placed  8  feet  from  the  base  of  a  building 
40  feet  high,  just  reached  the  top,  how  far  must  it  be  placed 
from  the  base  of  the  buildinir  that  it  may  ronrh  a  point  10 
feet  from  the  top? 

66.  Mr.  A.  is  35  years  of  age  and  his  son  is  lU.  How 
soon  will  the  son  be  on^-half  the  age  of  the  father? 


362  MISCELLA>'EOUS    EXAMPLES. 

^  67.  A  person  in  purchasing  sugar  found  that  if  he  bought 
sugar  at  11  cents  he  would  lack  30  cents  of  having  money 
enough  to  pay  for  it,  so  he  bought  sugar  at  lOJ  cents  and 
liad  15  cents  left.     How  many  pounds  did  he  buy? 

68.  A  farmer  had  his  sheep  in  three  fields.  J  ^^  ^^^  num- 
ber in  the  first  field  was  equal  to  f  of  the  number  in  the 
second  field,  and  |  of  the  number  in  the  second  field  was  J  of 
the  number  in  the  third  field.  If  the  entire  numlxjr  was  434, 
how  many  were  there  in  each  field  ? 

69.  A  and  B  can  do  a  piece  of  work  in  10  days,  B  and  C 
can  do  it  in  12  days,  and  A  and  C  in  15  days.  How  long 
will  it  take  each  to    do  it? 

70.  A,  B  and  C  pasture  an  equal  number  of  cattle  upon  a 
field  of  which  A  and  B  are  the  owners — A  of  9  acres  and  B 
of  15  acres.  If  C  pays  $24  for  his  pasturage,  how  much 
should  A  and  B  each  receive? 

71.  How  many  acres  are  there  in  a  square  tract  of  land 
containing  as  many  acres  as  there  are  boards  in  the  fence 
inclosing  it,  if  the  boards  are  11  feet  long  and  the  fence  is 
4  boards  high? 

72.  What  is  the  greatest  number  which  will  divide  27,  48, 
90,  and  174,  and  leave  the  same  remainder  in  each  case? 

73.  A  and  B  invested  equal  sums  in  business.  A  gained 
a  sum  equal  to  25^  of  his  stock,  and  B  lost  $225.  A's 
money  at  this  time  was  double  that  of  B's.  AVhat  amount 
did  each  invest? 

74.  A  man  at  his  marriage  agreed  that  if  at  his  death  he 
should  leave  only  a  daughter,  his  wife  should  have  f  of  his 
estate;  and  if  he  should  leave  only  a  son,  she  should  have  \. 
He  left  a  son  and  a  daughter.  What  fractional  part  of  the 
estate  should  each  receive,  and  how  much  was  each  one's 
portion,  if  the  estate  was  worth  86591  ? 


TEST   QUESTIONS.  363 


TEST  QUESTIONS. 

612.  Define  a  unit;  a  number.  Explain  the  necessity  for  a  uni- 
form system  of  grouping  objects.  In  how  many  ways  may  numbeni 
be  represented?  Name  them.  Define  numeration;  notation  ;  Arabic 
notation;  Roman  notation.  Give  the  first  principle  of  Arabic  nota- 
tion. Illustrate  it.  What  is  meant  by  "units  of  first  order,"  etc.? 
Give  the  general  principles  of  Arabic  notation.  What  is  meant  by 
a  {K'riod  of  figures?  Give  the  names  of  the  first  seven  periods.  Give 
the  rule  for  notation ;  for  numeration.  State  how  cents  and  mills 
are  written  in  notation  of  U.  S.  money.  What  characters  are  em- 
ployed in  Roman  notation?    Give  the  principles  of  Roman  notation. 

Define  addition;  sum,  or  amount;  equation;  like  numbers.  De- 
s<Tibe  the  sign  of  addition;  the  sign  of  equality.  How  many  cases 
are  there  in  addition?  Show  the  truth  of  the  principles  of  addition. 
RejHjat  the  rule  for  addition.  Why  do  we  begin  at  the  right  to  add? 
Why  are  the  numbers  of  the  same  order  written  in  the  same  column? 

Define  subtraction;  minuend;  subtrahend;  remainder;  diflTercncc. 
What  is  the  sign  of  subtraction?  What  is  it  called?  State  the  prin- 
ciples of  subtraction.  Show  that  they  are  true.  Explain  what  is  to 
be  done  when  some  figure  of  the  subtrahend  expresses  more  than  the 
corresfwnding  figure  of  the  minuend. 

IX'fine  multiplication;  multiplicand;  multiplier;  product;  factors 
of  the  multiplier ;  abstract  number.  Describe  the  sign  of  multipli- 
cation. Give  the  principles  of  multiplication.  Show  that  they  are 
true.  Show  that  multiplication  is  a  special  case  of  addition.  Repeat 
the  rule  for  multiplication.  What  steps  in  the  process  are  for  con- 
venience? How  may  you  multiply  when  there  are  ciphers  on  the 
right  of  cither  or  both  factors? 

Define  division;  dividend;  divisor;  quotient;  remainder.  What  is 
the  sign  of  division?  In  how  many  ways  is  division  indicated? 
State  the  principles  of  division.  Show  that  they  are  true.  Show  that 
division  is  a  special  case  of  subtraction.  In  how  many  ways  may 
the  remainder  l)e  expresse<l?  Hlustrate  each  way  by  an  example. 
What  is  a  fraction?  What  is  meant  by  long  division?  What  is 
meant  by  short  division?  Which  should  precede  the  other?  Why? 
What  steps  in  the  prooesj*  of  division  are  for  ronvcnience?  What  are 
necessary?     How   may  you  proceed  when  there  are  ciphers  on  the 


364  TEST   QUESTIONS. 

right  of  cither  divisor  or  dividend?  State  the  principlefl  governing 
the  relation  of  dividend,  divisor,  and  quotient.  Illustrate  each  by 
an  example.  Define  analysis.  Illustrate  the  process.  Describe  the 
parenthesis  and  vinculum,  and  show  their  uses. 

Define  and  illustrate  what  is  meant  by  an  integer;  exact  divisor; 
factor;  a  prime  number;  a  composite  number;  an  even  number;  an 
odd  number.  Give  eleven  facts  relating  to  exact  divisibility  of  num- 
bers. Illustrate  each  statement  by  an  appropriate  example.  What  is 
meant  by  factoring?  Prime  factors?  What  is  an  exponent?  State  the 
principles  relating  to  the  prime  factors  of  numbers.  Illustrate  the 
truth  of  these  principles  by  appropriate  examples.  Give  the  rule  for 
finding  the  prime  factors  of  a  number.  Explain  the  process  of  multi- 
plying by  factors.  Show  the  use  of  this  process.  Show  how  to  divide 
by  factors.  Explain  how  to  find  the  true  remainder  in  division  by 
factors.    Give  the  rule  for  dividing  by  the  factors  of  a  number. 

What  is  meant  by  cancellation?  Upon  what  principle  is  the  pro- 
cess based?     Illustrate  the  process. 

Define  what  is  meant  by  a  common  divisor;  the  greatest  common 
divisor;  numbers  that  are  prime  to  each  other.  What  is  the  princi- 
ple underlying  the  greatest  common  divisor?  Give  the  ordinary 
method  of  finding  the  greatest  common  divisor  when  the  numbers 
are  small.  Solve  an  example,  and  give  the  analysis  when  the  num- 
bers can  not  be  readily  factored. 

What  is  a  multiple?  Define  what  is  meant  by  a  common  multi- 
ple; the  least  common  multiple.  State  the  principle  upon  which  the 
processes  in  least  common  multiple  are  based.  Solve  an  example 
showing  the  truth  of  the  principle. 

Define  and  illustrate  what  is  meant  by  the  terms  fraction;  unit  of  a 
fraction;  fractional  unit;  the  denominator;  the  numerator;  thetermsof 
a  fraction;  a  proper  fraction;  an  improper  fraction;  a  mixed  number; 
a  common  fraction;  a  decimal  fraction.  How  are  fractional  expres- 
sions read?     Interpret  the  expression  f. 

WTiat  is  meant  by  reduction  of  fractions?  What  is  Case  I?  When 
is  a  fraction  reduced  to  larger  or  higher  terms?  Upon  what  princi- 
ple does  the  process  in  Case  I  depend?  What  is  Case  II?  What  is 
meant  by  reducing  a  fraction  to  smaller  or  lower  terms?  To  smallest 
or  lowest  terms?  Upon  what  principle  is  the  process  in  Case  II 
based?  What  is  Case  III  in  reduction?  Solve  an  example  illustrat- 
ing the  process.     W^hat  is  Case  IV?    Solve  an  example  illustrating 


TEST   QUEtoTiUN.S.  365 

the  process.  What  is  meant  by  similar  fractions?  Dissimilar  frac- 
tions? When  have  similar  fractions  their  least  common  denominator? 
Give  the  principles  relating  to  the  common  and.  least  common  denom- 
inator of  fractions.  What  is  the  rule  for  finding  the  least  common 
denominator  of  several  fractions? 

What  kind  of  fractions  only  can  be  added?  Why?  What  must 
be  done  with  dis.similar  fractions  before  they  can  be  added?  How 
should  mixed  numbers  Ikj  added?  W^hat  kind  of  fractions  only  can 
Ik?  subtracted?  What  must  be  done  to  dissimilar  fractions  before  they 
can  be  subtracted?     How  could  mixed    numbers  be  subtracted? 

What  is  Case  I  in  multiplication  of  fractions?  What  principle  un- 
derlies the  process?  Demonstrate  the  truth  of  the  principle.  What 
is  Case  II?  What  is  the  principle?  What  is  Case  III?  What  is  the 
general  rule  for  multiplication  of  fractions?  Solve  and  explain  the 
following:  Multiply  |  by  i. 

What  is  Case  I  in  division  of  fractions?  What  principle  underlies 
the  process?  Show  by  an  example  that  the  principle  is  true.  What 
is  Case  II?  Give  the  rule  for  dividing  an  integer  by  a  fraction. 
What  is  Case  III?  Solve  the  following:  What  is  the  value  of  ^-f-|? 
Give  an  analysis  and  explanation  of  the  process.  Give  the  general 
rule  for  division  of  fractions.  Dc?cril)e  what  arc  included  among 
fractional  forms  How  arc  they  simplified?  What  is  Case  I  in  frac- 
tional relation  of  numbers?  What  is  the  principle  upon  which  rela- 
tion of  numbers  is  based?  What  is  Case  II?  Illustrate  each  case  by 
an  example. 

What  is  a  decimal  fraction?  From  what  is  the  word  decimal 
derived?  How  are  decimal  fractions  expressed?  How  are  decimals 
distinguished  from  integers?  State  the  principles  of  decimal  frac- 
tions. Show  each  to  be  true.  What  is  the  decimal  point?  What 
other  name  has  it?  What  is  a  pure  decimal?  What  is  a  mixed  deci- 
mal? What  is  a  complex  decimal?  Name  the  orders  of  dcrimals  as 
far  as  ten-millionths.  How  docs  the  place  occupied  by  any  order  of 
decimals  compare  with  that  occupied  by  integers  of  the  correspond- 
ing nan»e? 

How  are  decimals  reduced  to  a  common  denominator?  Explain 
the  prcKH'ss.  How  are  common  fractions  reduced  to  decimals? 
Analyze  the  protxss.  If  a  common  fraction  can  not  be  exactly  re- 
duced to  a  decimal,  what  is  done?  How  do  addition  and  subtrnction 
of  decimals  compare  with  the  same  processes  in  integers. 


360  TEST   QUESTIONS. 

Wljat  is  the  principle  upon  which  multiplication  of  docinials  is 
ba«ed?  Show  that  it  i.s  true.  How  may  a  decimal  be  multiplied  by 
1  with  any  number  of  ciphers  annexed?  What  is  the  principle  uiwn 
which  the  process  of  division  of  decimals  is  based?  How  may  a  deci- 
mal be  divided  by  1  with  any  number  of  ciphers  annexed? 

How  may  we  multiply  by  a  number  that  is  a  little  less  than  u  unit 
of  the  next  higher  order?  How  may  we  multiply  when  one  part  of 
the  multiplier  is  a  factor  of  another  part?  How  may  we  multii)ly  by 
a  numl)er  that  is  a  part  of  some  higher  unit?  What  is  an  aliquot  part 
of  a  number?  What  are  the  common  aliquot  parts  of  10?  What  of 
100?  How  is  the  cost  found  when  Ihc  (lunntity  and  price  jkt  100  or 
1000  are  given? 

What  is  a  debt?  Define  wUai  i-  i..v..;u  by  a  credit;  a  debtor;  a 
cretUtor;  an  account;  the  balance  of  an  account;  a  bill;  the  footing 
of  a  bill.  State  some  of  the  more  common  abbreviations  used  in 
business  correspondence. 

Tell  what  a  concrete  number  is;  an  abstract  number;  a  denominate 
number;  a  simple  denominate  number;  a  comiK)und  denominate  num- 
ber; a  standard  unit;  a  scale.  Illustrate  each  of  the  preceding  by  an 
appropriate  example.    How  many  kinds  of  numerical  scales  are  there? 

What  is  money?  Of  how  many  kinds  is  it?  What  i^  coin,  or 
specie?  W^hat  is  paper  money?  Give  the  table  and  denominations 
of  the  currency  of  the  United  States.  What  are  the  ordinary  coins? 
What  are  the  denominations  and  coins  of  Canada?  Give  the  table 
of  English  money  and*  the  coins  in  common  use.  What  are  the  cur- 
rency and  coins  of  France? 

What  is  meant  by  reduction  of  denominate  numbers?  What  is 
reduction  descending?  Give  the  rule.  What  is  reduction  ascending? 
Give  the  rule. 

Detine  and  illustrate  what  is  meant  by  space,  a  line,  a  surface,  a 
solid.  For  what  are  linear  measures  used?  Repeat  the  table  of 
Linear  Measure,  and  of  Surveyor's  Linear  Measure.  What  is  an 
angle?  A  square?  A  square  inch?  A  rectangle?  Wiiat  is  the  area 
of  a  surface'  How  is  the  area  of  a  rectangular  surface  computed? 
Repeat  the  table  of  Square  Measure,  and  of  Surveyors'  Square  Meas- 
ure. What  is  a  solid?  A  cube?  A  cubic  inch?  A  cubic  foot?  The 
volume  or  solid  contents?  How  is  the  volume  of  a  rectangular  solid 
computed?  Repeat  the  tables  of  Cubic  Measure,  and  Wood  and  Stone 
Measure. 


TEST   QUESTIONS.  367 

What  are  the  measures  of  capacity?  Recite  tlie  table  of  Liquid 
Measure.  In  estimating  the  capacity  of  cisterns,  etc.,  how  many  gal- 
lons are  considered  a  barrel?  How  many  a  liogshead?  How  many 
cubic  inches  are  there  in  a  gallon?  Repeat  the  table  of  Apothecaries'. 
Fluid  Measure.  For  what  is  Dry  Measure  used?  Repeat  the  table. 
How  many  cubic  inches  are  there  in  a  bushel? 

What  is  weight?  For  what  is  Avoirdupois  Weight  used?  Repeat 
the  table.  How  many  pounds  are  there  in  the  long  ton?  How  many 
grains  are  there  in  an  avoirdupois  pound?  For  what  is  Troy  Weight 
U8e<l?  liei)eat  the  table.  How  many  grains  are  there  in  a  Troy 
jwund?  For  what  is  Ajwthecaries'  Weight  used?  Repeat  the  table. 
How  many  grains  are  there  in  a  pound  Apothecaries'  Weight? 

Repeat  the  table  of  Measures  of  Time.  Explain  how  often  leap 
year  occurs.  What  is  a  circle?  What  is  the  circumference  of  a  circle? 
An  arc  of  a  circle?  A  degree  of  the  circumference?  What  is  the 
measure  of  an  angle?  Repeat  the  table  of  Circular  Measure.  What 
a  quadrant?  A  se.xtant?  Give  the  Stationers'  Table  and  the  table 
of  Counting.  Give  the  cases  in  lieduction  of  Denominate  Fractions. 
Solve  an  example  illustrative  of  each  case  and  explain  the  process. 

How  do  the  fundamental  processes  in  Compound  Denominate  Num- 
bers compare  with  the  same  proces.ses  in  Simple  Numbers? 

How  d(H"s  the  number  of  degrees  apparently  passed  over  by  the  sun 
compare  with  the  number  of  hours  occupied  in  passing  that  distance? 
The  number  of  minutes  of  space  with  the  number  of  minutes  of  time? 
The  seconds  of  space  with  the  sectmds  of  time?  Iie|>eat  the  table 
showing  the  relation  between  longitude  and  time.  What  is  a  merid- 
ian? What  is  longitude?  Give  the  rule  for  Hnding  the  difference  in 
time  when  the  difference  in  longitude  of  two  places  is  given.  Give  the 
rule  for  finding  the  difference  in  longitude  of  two  places  when  their 
difference  in  time  is  given. 

What  is  the  unit  of  length  in  the  Metric  System  of  measures?  To 
what  is  it  nearly  e<iual?  What  is  the  metric  unit  of  area?  What 
the  unit  of  solidity?  What  the  unit  of  capacity?  What  the  unit  of 
weight? 

Define  jK?r  cent.  What  is  the  commercial  sign  of  per  cent.?  Of 
what  d(H'8  Percentage  treat?  How  may  per  cent,  be  expressetl? 
What  i-lemcnts  are  involve<l  in  problems  in  Percentage?  What  it) 
meant  by  the  basi'?  The  rate?  The  |»ercentage?  The  amount?  The 
difference?     What  arc  the  five  fundamental   problems  or  cases  in 


368  TEST   QUESTIONS. 

Percentage?  Solve  an  examyl"  iMnctmtinrr  n->oh^  nnd  givo  u  rulo  for 
each  case. 

What  is  interest?  Define  mo  tirins  i>rMi(ii)aI ;  amount;  rate  of 
interest;  legal  interest;  usury;  a  note  or  promissory  note.  Give  three 
methods  for  computing  interest.  What  is  compound  interecl?  Give 
the  rule  for  computing  compound  interest.  How  is  the  com^Mjund 
interest  table  formed?  What  is  meant  by  annual  interest?  Give  the 
rule  for  computing  annual  interest.  In  wh.'t  nspeot  dfx^s  ronnH.imd 
interest  differ  from  annual  interest? 

What  are  partial  payments?  WMiat  is  an  indorsement.''  \\ii;ti  is 
the  Mercantile  Rule  for  computing  the  amount  due  when  partial  pay- 
ments have  been  made?  When  is  the  Mercantile  Rule  used?  What 
is  the  principle  upon  which  the  United  States  Rule  is  based?  Give 
the  United  States  Rule.  When  the  principal,  rate,  and  interest  are 
given,  how  is  the  time  found?  When  the  principal,  time,  and  interest 
are  given,  how  is  the  rate  found?  When  the  rate,  time,  and  interest 
are  given,  how  is  the  principal  found? 

What  is  a  promissory  note?  Wiio  is  the  maker  or  drawer?  Who 
is  the  payee?  Who  is  the  holder?  Who  is  the  indorser?  In  how 
many  ways  may  he  indorsa?  Wiiat  is  the  face  of  a  note?  W^iien  is  a 
note  negotiable?  When  is  a  note  not  negotiable?  What  are  days  of 
grace?     Write  a  negotiable  note  and  transfer  it  by  indorsement. 

What  is  discount?  What  is  commercial  discount?  What  is  net 
price?  What  is  the  cash  value  of  a  bill?  In  cases  where  there  is  a 
discount  of  some  per  cent.,  as  20^/c  oflf  and  o^c  off  for  cash,  upon  what 
sum  is  the  5^  computed?  What  is  true  discount?  Define  present 
worth.  Give  the  rule  for  solving  problems  in  true  discount.  W^hat  is 
a  bank?  A  check?  Bank  discount?  The  proceeds  or  avails  of  a 
note?  The  maturity  of  a  note?  The  term  of  discount?  How  is  the 
bank  discount  computed?  Is  it  right  or  wrong  in  principle?  How 
can  we  find  how  large  to  make  a  note  that  we  may  have  a  certain  sum 
left  after  paying  the  discount  at  a  bank? 

W^hat  elements  in  Profit  and  Loss  correspond  to  the  base,  rate,  per- 
centage, amount,  and  difference?  What  is  the  principle  upon  which 
computations  in  Profit  and  I^ss  are  based? 

Define  the  terras  commission  merchant  or  agent;  commission;  a  con- 
signment; consignor;  consignee;  the  net  proceeds.  What  elements  in 
commission  correspond  to  base,  rate,  percentage,  amount,  and  differ- 
ence?   Upon  what  principle  is  commission  based? 


TEST   QUESTIONS.  369 

AVliat  is  u  tax?  What  is  real  estate?  What  is  personal  proiKrty? 
What  is  a  property  tax?  What  is  a  personal  tax?  Who  is  an 
assessor?  .What  is  an  assessment  roll?  Explain  how  taxes  are 
levied  and  the  individual  taxes  computed.  Exphiin  the  formation  of 
the  assessor's  table.  What  are  duties  or  customs?  What  is  meant  by 
P{)ecific  duty?  Ad  valorem  duty?  Tare?  Leakage  and  breakage? 
Custom-houses? 

What  is  meant  by  the  terms,  a  company ;  a  corporate  company  or 
corporation;  a  charter;  capital  stock;  a  share  of  stock;  a  certificate  of 
stock;  })ar  value;  above  par;  below  par;  market  value?  What  is  an 
installment?  What  is  an  assessment?  What  is  a  dividend?  De- 
scribe a  bond  and  coupons.  How  are  Government  securities  desig- 
nated? Name  the  various  classes  of  Government  securities,  and  state 
the  rate  of  interest  they  bear.  In  what  are  all  Government  bonds 
payable?  In  what  is  the  interest  of  all  Government  bonds  payable? 
W^hat  are  stocks?  "Who  is  a  stock-broker?  What  is  brokerage? 
Wljat  elements  in  the  subject  of  stocks  correspond  to  base,  percentage, 
amount,  and  difference?  What  is  meant  by  tlie  expressions,  stock  is 
selling  at  83 [,  112,  etc.?  If  the  market  value  and  rate  of  premium  or 
discount  are  giver,  how  can  the  par  value  be  found? 

What  is  insurance?  Of  how  many  kinds  is  it?  What  is  property 
insurance?  WHiat  is  a  policy?  W^hat  is  the  premium?  Of  how 
many  kinds  are  insurance  companies  with  regard  to  the  parties  who 
participate  in  the  profits?  What  is  a  mutual  insurance  company? 
What  is  a  stock  company?  What  is  a  mixed  company?  What  are 
the  elements  involved  in  insurance  that  correspond  to  base,  rate,  and 
l>erccntagc?  What  is  personal  insurance?  W^hat  is  a  life  policy? 
What  an  endowment  policy?    W^hat  an  accident  or  health  policy? 

What  is  exchange?  Explain  the  method.  WMiat  is  a  draft  or  bill 
of  exchange  ?  Write  a  draft.  How  many  parties  are  there  connected 
with  a  draft  primarily?  WHio  is  the  drawer?  The  drawee?  The 
payee?  What  is  a  sight  draft?  A  time  draft?  What  is  meant  by 
accepting  a  draft?  Of  how  many  kinds  is  exchange?  What  is  do- 
mestic exchange?  State  and  solve  examples  illustrating  the  rules. 
What  is  foreign  exchange?  "Wliat  is  a  sot  of  exchange?  Upon  what 
cities  in  Europe  are  drafts  more  commonly  drawn?  What  is  the 
value  in  United  States  gold  coin  of  a  stivereign?  Wliat  is  the  value 
of  a  franc?  P*'-'  -i  rmmple  ilhicfratinir  the  principles  of  foreign 
exchange. 
24 


370  TEST  QUESTIONS. 

What  18  meant  by  averaging  payments?  The  avcrai,'i!  linic .'  Tlio 
term  of  credit?  The  average  term  of  cre<iit?  Solve  an  example  illus- 
trating the  proce«H  of  solution  when  the  term«  of  credit  begin  at  the 
same  time.  Solve  an  example  when  the  terms  of  cretlit  begin  at  dif- 
ferent dates.  Explain  the  process.  Explain  the  process  of  averaging 
accounts,  and  give  the  rule. 

What  is  partnership?  Who  are  partners?  Wiiat  is  the  capital  of 
a  firm  or  company?  Give  the  principle  underlying  partnership. 
What  is  Case  I  in  partnership?  What  is  Case  II?  Solve  and  analyze 
an  example  in  Case  II.    Give  the  rule  for  partnership  settlements. 

What  is  ratio?  Of  how  many  kinds  is  it?  What  are  the  terms  of 
a  ratio?  The  antecedent?  The  consequent?  What  is  the  sign  of 
ratio?  From  what  may  it  be  regarded  as  derived?  What  is  a  coup- 
let?   What  are  the  principles  of  ratio?     Illustrate  each  principle. 

What  is  proportion?  What  is  the  sign  of  proportion?  From  what 
may  it  be  regarded  as  derived?  What  are  the  antecedents  of  a  pro- 
portion? The  consequents?  The  extremes?  The  means?  Give  the 
principles  of  pro|>ortion  regarding  the  relation  of  extremes  and  means. 
W^hat  is  meant  by  a  simple  ratio?  A  simple  proportion?  A  direct 
proportion?  An  inverse  proportion?  Give  the  rule  for  solving  ex- 
amples in  simple  proportion.  What  is  a  com}K)und  ratio?  A  com- 
pound proportion?  The  principle  underlying  compound  proportion? 
Solve  an  example  in  compound  projKjrtion,  by  successive  simple  pro- 
portions and  by  cause  and  cflect.  Give  the  rule  for  compound  pro- 
portion. 

W^hat  is  a  power?  How  are  powers  named?  What  is  an  exponent? 
What  is  involution?  How  is  the  power  of  a  number  obtained?  How  fl 
is  the  square  of  a  number  expressed  in  terms  of  its  tens  and  units? 
How  in  terms  of  any  two  parts?  How  is  the  cube  of  a  number  ex- 
pressed in  terms  of  its  tens  and  units?  How  in  terms  of  any  two 
parts? 

W^hat  is  a  root?  How  are  roots  named?  W^hat  is  evolution? 
What  is  the  radical  or  root  sign?  Define  a  perfect  power;  an  im- 
perfect power.  Give  the  rule  for  finding  the  root  of  a  number  by 
factoring.  Give  the  principle  relating  to  the  number  of  figures  re- 
quired to  express  the  square  of  a  number;  the  cube  of  a  number. 
Give  the  principle  relating  to  the  number  of  figures  in  the  square 
root  of  a  number;  the  cube  root  of  a  number.  Solve  and  explain  an 
example  in  square  root.     Repeat  the  rule.    What  is  done  when  the 


TEST   QUESTIONS.  371 

nuiiilxT  is  not  a  perfect  square?  How  are  decimals  pointcil  oil? 
How  is  the  square  root  of  a  common  fraction  found?  How  is  the  side 
of  a  stjuare  found  when  its  area  is  given? 

What  relation  do  the  squares  described  u|)on  tlic  sides  of  a  right- 
angled  triangle  sustain  to  each  other?  How  is  the  hypotenuse  of  a 
right-angled  triangle  found  when  the  other  sides  arc  given?  How  is 
eitlier  side  found  when  the  hypotenuse  and  the  other  side  are  given? 
What  are  similar  figures?  What  is  the  relation  between  similar 
surfaces  ? 

Solve  an  example  in  cube  root.  Deduce  from  your  solution  a  rule. 
What  is  done  when  there  is  a  remainder  after  subtracting  tlie  last 
product?  How  are  decimals  pointed  ofT?  How  is  the  cube  root  of 
a  common  fraction  found?  Give  the  principles  relating  to  similar 
solids. 

Define  a  series;  an  ascending  series;  a  descending  scries;  an  arith- 
metical progression;  a  geometrical  progression.  How  is  the  sum  of 
an  aritmetie'al  series  found?  Illustrate  by  an  example.  How  is  the 
sum  of  a  geometrical  series  found?     Illustrate  by  an  example. 

Define  mensuration;  a  line;  a  straight  line;  a  curved  line;  paral- 
lel lines;  a  plane  surface;  a  curved  surface;  an  angle;  a  right  angle; 
an  acute  angle;  an  obtuse  angle;  a  vertex  of  an  angle;  a  triangle; 
a  <juadrilateral;  a  parallelogram;  a  rectangle;  a  polygon;  a  circle; 
the  circumference  of  a  circle;  a  radius  of  a  circle;  a  diameter  of  a 
circle;  the  base  of  a  figure;  the  altitude  of  a  figure;  a  diagonal  of  a 
figure;  the  "perimeter  of  a  figure;  the  area  of  a  surface. 

How  is  the  circumference  of  a  circle  obtained  from  its  diameter? 
The  diameter  from  the  circumference?  How  is  the  area  of  a  paral- 
lelogram computed?  How  is  the  area  of  a  triangle  computed?  Give 
the  rule  for  computing  the  area  of  a  trapezium;  a  trapezoid;  a  n  lth- 
lar  polygon;  a  circle. 

What  is  a  solid?  A  prism?  A  parallelopipedon?  A  cylinder? 
A  pyramid?  A  cone?  A  frustum?  A  sphere?  A  diameter?  A 
radius  of  a  sphere?  The  circumference  of  a  sphere?  The  altitude 
of  a  solid?    The  convex  surface  of  a  solid? 

Give  the  rule  for  finding  the  convex  surface  of  a  prism  or  cylinder; 
a  pyraniid  or  cone;  a  frustum  of  a  pyramitl  or  e*one;  a  sphere.  Give 
the  rule  for  finding  the  volume  of  a  prism  or  cylinder;  a  pyramid  or 

cone-     •(     ffi^t.....    ..f    ..     ■>vr..i..;,l    ,>r    ,.,.1....     ..    i;i.lw.r,. 


^'answers 


r 


PllS«2S. 

2.  947. 

3.  1*498. 

4.  S42.70. 

5.  $28.48. 
C.  10845. 

7.  20839. 

8.  6283. 

9.  3530. 

10.  21974. 

11.  20002. 

12.  $10:vi-\ 

13.  sh;;:  M>. 

14.  U33;i24. 

15.  134739. 

16.  349950638. 

17.  $132.94. 

18.  1600. 

Pagrc  29. 

19.  $14593. 

20.  21502. 

21.  9265. 

22.  2775. 

23.  8692. 

24.  934. 

25.  25917. 

Pa$rc  30. 

20.  997.  ' 

27.  645621. 

28.  528408. 

29.  $8484. 

30.  7005  lib.; 
1999881  vol. 

31.  402399. 

32.  377805. 

33.  642502. 

(372) 


PflS«  SI.               1 

18.  182. 

34.  42151. 

19.  1S9. 

35.  118422. 

20.  367. 

36.  12149865. 

21.  180. 

37.  $5165000. 

22.  93. 

38.  5871  ixTiod. 

23.  865. 

20842475  circ. 

24.  3886. 

25.  5828. 

P»«e33. 

26.  2131. 

39.  $1595.85, Albany; 

27.  8132. 

$4807.37,  N.Y.; 

28.  10570. 

$()403.22,  both. 

40.  45704. 

41.  47388. 

42.  40590. 

29.  20068. 

30.  28879. 

31.  80091. 

32.  8486888. 

43.  540356. 

33.  256  mi. 

44.  8882. 

34.  $508. 

45.  11031. 

46.  109546. 

Pngrc  42, 

47.  86672. 

35.  11 63-. 

30.  SI  31. 

Paire41. 

37.  1955. 

2.  305. 

38.  009973105. 

3.  228. 

39.  159  acres. 

4.  292. 

40.  S202.12. 

5.  272. 

41.  1492. 

6.  879. 

42.  $1097. 

7.  61. 

43.  S9036. 

8.  919. 

44.  8087. 

9.  288. 

45.  3502. 

10.  1299. 

11.  150. 

Page  48 

12.  $3.07. 

2.  942. 

13.  $11.26. 

3.  2272. 

14.  SI  6.07. 

4.  3948. 

15.  S23.96. 

5.  1725. 

16.  $23.67. 

6.  $4095. 

17.  289. 

7.  $2307. 

ANSWERS. 


73 


8.  1570. 

9.  2(>20. 

10.  b'MS. 

11.  24U0. 

12.  4404. 

13.  $432. 

14.  $.330. 

15.  S192. 
10.  S390. 

17.  $138. 

18.  $10555. 

19.  "^^3.75. 

20.  $250.10. 

Pns(>  49. 

21.  3825. 

22.  1508. 

23.  $10.02. 

24.  $43.75. 

25.  3()900. 
20.  $:i00. 

27.  $127.75. 

28.  $008. 

Piigre  53. 

2.  i:i050. 

3.  11S20. 

4.  I7r,(;4. 

5.  $<;4.03. 
0.  $114.48. 

7.  5472. 

8.  8947. 

9.  12325. 

10.  1009S. 

11.  1(U.')(K). 

12.  74.S()3. 

13.  2311K)3. 

14.  34.S.5092. 

15.  3'.  M  J 140. 
10.  <i5(i.S848. 

17.  1344456. 

18.  i:W2075. 

19.  .585575. 

20.  157W.88. 

21.  .3334932. 

22.  254()(J50. 

23.  23748222. 

24.  25721944. 

25.  02748374. 


20.  47913214. 

27.  00319209. 

28.  92317008. 

29.  70520998. 

30.  82013817. 

31.  14.%55004. 

32.  291000415. 

33.  281072720. 

34.  1151707808. 

35.  570873555. 
30.  1381000350. 

37.  1251750030. 

38.  1253542212. 

Pa^rc  03. 

39.  $1368.53. 

40.  $5211.46. 

41.  $13309.92. 

42.  $99253.80. 

43.  $152323.35. 

44.  $2,332.99. 

45.  $09520.33. 
40.  $09577.43. 

47.  $213409.50. 

48.  $202810.80. 

49.  58050. 

60.  11024. 

61.  200520. 

62.  6022. 
53.  $48635. 
64.  $35105. 
55.  ^108000. 

Pnfgo  5t. 

3.  3750; 
37500; 
15000; 
112500. 

4.  26350; 
59150; 
507000 ; 
7(J05(M>. 

5.  88000; 
12:i200; 
70400; 
]7<WK'il. 
2771  M)(l; 
277U(M>(»; 
3047000; 


0.  27: 


4570500. 

7. 

1215000; 

15662000; 

20412000; 

31590000. 

8. 

055200 ; 

11850000; 

8424000; 

14362000. 

Pagre  55. 

9. 

2040000. 

10. 

48000. 

11. 

$832. 

12. 

$18256. 

1. 

$32739.84. 

2. 

$13730.50. 

3. 

1229. 

4. 

339904. 

5. 

$4299. 

6. 

$108. 

7. 

10293. 

8. 

4770. 

9. 

$35.90. 

I»ng:e  56. 

10. 

89232. 

11. 

$247.52. 

12.  $169.25. 

13. 

10248. 

14. 

108  mi. 

16. 

$636.14. 

16. 

$17481.30. 

17. 

$2000.95. 

18. 

$1503.25. 

Pngrc  64. 

11. 

1218. 

12. 

1360. 

13. 

496. 

14. 

597. 

15. 

1545. 

16. 

984. 

17. 

400. 

18. 

876. 

19. 

554. 

20. 

1:J53. 

21. 

548. 

874 


ANSWERS. 


22.  1234. 
2:}.  913?. 

24.  1488  J. 

25.  9081. 

26.  2296k 

27.  12051. 

28.  485|. 

Pace  67. 

2.  124. 

3.  318. 

4.  216. 

5.  306. 

6.  406. 

7.  432. 

8.  416. 

9.  46. 

10.  312. 

11.  442. 

12.  41. 

13.  22. 

14.  77. 

15.  103. 

16.  213. 

17.  119. 

18.  131. 

19.  114. 

20.  141. 

21.  67. 

22.  505. 

23.  315. 

24.  406. 

25.  519. 

26.  525. 

27.  541. 

28.  606. 

29.  723. 

30.  544. 

31.  752. 

32.  664. 

33.  777. 

34.  1802. 

35.  1945. 

36.  4372. 

37.  0203. 

38.  9216^]  ^ 

39.  5()79?^.«. 

40.  12474^  ^^ 

41.  2117o,^^5. 


42.  81987/^y. 

43.  771849}!?. 

44.  474636M'y. 

45.  252384HH. 

46.  12752§jHt. 

47.  854104|?|*|. 

Pmc«  6S, 

48.  162. 

49.  $258. 

50.  130  mi. 

51.  18  doz. 

52.  1093f9fC. 

53.  5nsf. 

54.  140  da.  480  left 

56.  $2029J. 

57.  $l0086jV 

58.  5280. 

59.  100.  j 
GO.  21jft/»M. 

Pllff«70.  ' 

3.  i8yy\,. 

4.  6liJ^ 

5.  127«a. 

6.  39|H. 

l-M. 

0.  75^^^. 

10.  50^5V(T. 

11.  779HM*.. 

12.  118;  4964  bu.  rem. 

13.  640. 

IPnge  73. 

19.  2546?. 

20.  S42m. 

21.  8Wj'J>,  wk. 

22.  $60. 

23.  1920  bu. 

24.  $2  per  bbl. 

25.  45  vd. 

26.  500'  lui. 

27.  20  da. 

28.  80  bu. 
29    60  acres ; 

$400  loss. 


30.  $38,37.50. 

31.  $92278if2f. 

32.  18  minutes. 

33.  105  head. 

34.  lOgrandchild'n. 

35.  3716033  lb. 

36.  21  HiJc  acres. 

rage  79. 

2.  7,  5. 

3.  2«. 

4.  2*  3.  7. 


5.  2' 

6.  2* 

7.  3^ 

8.  2, 

9.  2» 

10.  2, 

11.  22 

12.  22 

13.  2» 

14.  2* 

15.  22 
10.  2* 

17.  2* 

18.  22 

19.  2, 

20.  22 

21.  33 

22.  7, 

23.  22 

24.  72 

25.  22 

26.  2^ 

27.  22 

28.  22 

29.  23 

30.  22 

31.  2« 


3,7. 
32. 
5,7. 
32,  11. 
7. 

,131. 
79. 

112. 

5. 

32,7. 

3,  5,  19. 

37. 

32,  13. 

3,  72,  13. 
3,5,7,11. 

32,  89. 

52,7. 
13,  43. 

953. 

11,  13. 

5,  199. 

5,11,61. 

3,  11,  172. 

S\  5,  7. 

3,7,11,  13. 

5?,  37. 

3\ 


Pagt*  80. 

6.  13600; 
15300; 
20400; 
30000. 

7.  102144; 
49248; 
82080; 
196992. 


ANSWKllS. 

8.  $2240. 

Pag©  86. 

4.  630. 

9.  $112.35. 

2.  4. 

5.  144. 

10.  ?rj*t<m. 

3.  9. 

6.  300. 

11.  S-'J  '. 

4.  12. 

7.  540. 

12.  ^I'-Jos. 

5.  6. 

8.  770. 

i.i.  $i;i20. 

/>.  12. 

9.  720. 

14.  $1620. 

7.  9. 

10.  960. 

15.  $131.00. 

8.  16. 

11.  2835. 

9.  9. 

12.  6650. 

VHge  81. 

10.  7. 

13.  288. 

9.  71. 

11.  8. 

14.  2240. 

10.  315. 

12.  12. 

15.  700. 

11.  205. 

13.  5. 

16.  41580. 

12.  34. 

14.  6. 

17.  401115. 

13.  23. 

14.  49. 

15.  7. 

16.  4. 

Page  91. 

15.  119i|. 

16.  44H. 

17.  58§J. 

18.  121/7. 

19.  91A. 

20.  3042V 

17.  11. 

18.  24. 

19.  360. 

20.  1620. 

19.  42. 

21.  240. 

Pagre  87. 

2.  13. 

22.  1440. 

23.  900. 

24.  85800. 

3.  11. 

25.  48. 

Pmre82. 

4.  12. 

26.  480. 

21.  224  canistere ; 

5.  17. 

6.  27. 

27.  1200. 

28.  20160. 

32  packages. 

7.  28. 

29.  240. 

J  J.  l')2  packages; 

8.  42. 

30.  24  inches. 

38  quires. 

9.  52. 

31.  60  yards. 

PKVeSS. 

10.  27. 

11.  21. 

32.  120. 

33.  210  bu. 

3.  7. 

12.  34. 

34.  360. 

4.  2. 

13.  4. 

5.  3. 

14.  32. 

Pas«1>a. 

6.  2|. 

15.  17.  * 

35.  120  days. 

7.  4. 

16.  33. 

36.  436A. 

8.  3?. 

17.  126. 

37.  12  feet. 

9.  4. 

18.  42. 

38.  4. 

10.  41. 

11.  Id. 

19.  37. 

39.  5276. 

20.  14  in. 

40.  1728. 

12.  186  J. 

41.  $336. 

PaiceSS. 

42.  2025. 

Fn««  84. 

21.  5  lbs. 

43.  SiKxk. 

13.  7. 

22.  67  fieldii; 

44.  41}. 

14.  10425. 

()  acres  eacli. 

45.  7. 

15.  107800. 

46.  378. 

16.  $43y. 

17.  7iJ  bu. 

2.  448. 

Pil««M. 

18.  $3.12. 

3.  144. 

2.  IJ. 

376 


376 


ANSWERS. 


Pace  lOOi 


Page  lot. 

6.  ¥;  If. 
8.  W;  W- 


Puce  lOS. 

2.  $5. 

3.  12, 

4.  101 

5.  16j 

6.  12j 

7.  21- 

9.'  25tV. 

10.  13H. 

11.  27i^ 

12.  15^,. 

13.  6H|. 

14.  18}. 

15.  14^. 

16.  ISjI. 

17.  12911  J.* 

18.  Si^^j. 

19.  58^*,. 

20.  97f|J. 

21.  16tV^. 

22.  6ffH. 

23.  3|^^|. 

P»gre  104. 

3.  M;  if ;  M. 

6.  i?g;  tVV;  1*^. 

7.  TV.;K'i;  t\V 

8.  '        "      ■  ■ 


■^1? 


J>.  1^;  }S?;  iVo. 
10.  fA;  HI;  in- 

i2  pi  lib  i^' 

14.  iU;  m;  m- 


Pace  10«. 


Pace  107. 

n.  2/A. 

12.  32}|. 

13.  41j 

14.  29[ 

15.  sm- 

16.  2^HH. 

17.  $246yV 

18.  $115'. 

19.  1431gi. 

20.  361^. 

Pnce  108. 


3.  U- 

4.1!. 

7.  ^i,. 


Page  109. 


9.  ^3^. 

10.  h 

11.  tV 

12.  /,. 

13.  H- 

14.  tItt. 

15.  lOi^i. 
IG.  33yV 

17.  100|-|. 

18.  36 1. 

19.  536^. 


20.  521ii. 

21.  88J. 

22.  S3. 

23.  222V  sold; 

189^  left. 

24.  S19.I. 

Page  llO. 

2.  lA. 

3.  4i. 

4.  If 

l:h. 

7.  6. 

8.  f 

9.  li 

■. 

10.  2 

.. 

11.  3 

12.  4 

f. 

13.  5 

14.  5A. 

16.  sn. 

16.  llA. 

Pase  111 

17.  $5J. 

18.  $5. 

20.  861. 

21.  $8lj. 

22.  154|. 

Page  112. 

2.  SI 

3.  2|. 

4.  4l. 

5.  Oj. 

6.  12. 

7.  99. 

8.  15. 

9.  24U. 

10.  $76/5. 

11.  $87. 

12.  $549^^. 

13.  $4218. 

14.  $4710. 

15.  88. 

17.  1 

Oft: 

ANSWERS. 

320A; 

Ji-  W- 

370. 

12.  :^. 

13.^. 

Page  114. 

2.  |t. 

Page  117. 

15.  2H. 

4!  i8- 

16.  6A. 

5.  >'. 

17.  4^. 

6.  ^5. 

18.  2^. 

7.  |. 

19.  3U. 

20.  3||. 

8.  I'f 

9.  1|. 

21.  i. 

10.  s'f. 

11.  Th- 

22.  7f,r. 

23.  $4H. 

12.  3j§i,r- 

24.  $|. 

13.  xk. 

14.  iV 

15.  a?. 

16.  U- 

25.  $11  ^V 

26.  $10723V 

Page  118. 

17.  ^V 

2.  42. 

18.  J>,. 

3.  72. 

19.  ? 

4.  111. 

20.  A. 

21.  U, 

5.  56. 

6.  80. 

22.  i. 

7.  86i. 

24.  3,V. 

8.  60. 

25.  27. 

9.  130. 

26.  31 1. 

10.  120. 

27.  147 

11.  401. 

12.  19}. 

28.  $?. 

29.  90». 

13.  93}. 

30.  2. 

14.  49. 

31.  634^ 

15.  68. 

32.  $81. 

16.  69. 

17.  123. 

Pave  115. 

18.  60. 

34:3% 

19.  111. 

20.  305. 

21.  145. 

Pas«  116. 

22.  128. 

tt 

23.  63t>j. 

24.  48^. 

25.  21^,. 

Page  119. 

7.x. 

27.  4f. 

8.5?. 

28.  Sj'j. 

9.  A- 

^-  Sit- 

10.  /,. 

30.  5||. 

;77 


378 


A^^\\  \:\>. 


1.  4H. 

\.  20  V 


d. 


31 
32 
33. 

34.  IM  ■  l)u 

35.  2U. 
3G.  $9^. 

37.  $/.. 

38.  f  Jf  lb. 


PMr«  i>i« 


S.f|. 


9. 

10.  if. 

11.  1. 

12.  f^. 

18.  ,V. 
10.  211 

20.  2A. 

21.  \l 

22.  7^. 

23.  3^ 

24.  201. 

25.  93|f. 

Pa9«  122. 

20.  46. 

27.  71. 

28.  82V. 

29.  1§. 

30.  13^1. 


31.  A. 

32.  1689  i 


bu. 


Pa«:e  133. 


2.  /i. 

5.  1^,. 

6.  A. 

It 

11.  2rh. 


12.  H. 

13.  llA. 


Pave  124. 


!*»««  125. 

13.  H;  V. 

4.  i;l|. 
15.  i,     I. 

16. 1;  |. 

1.  $6|. 

2.  S71.30. 

3.  2\  acres. 

Page  126. 

4.  64jV  cents. 

5.  6i. 

6.  $.17|f. 

7.  37i. 

8.  $925tVj  gain. 

9.  I63.V    ■ 

10.  2066fk 

11.  153k 

12.  9tV. 

13.  $2850,    sou; 
$2533],  daughter 


14.  $89}. 

15.  16  bu. 

16.  ii. 

Pii«e  12T. 

17.  222. 

18.  |6i|,. 

20.  A,A'8,or$4363l; 
|?.B'8,or|8726i. 

21.  $105. 

22.  $3U. 

23.  12i8. 

24.  $113541. 

25.  $15986}. 

26.  $24,Vr,  first; 
*28j'y^-,  second. 

27.  $2000,  one's; 
$3900,  other's. 

28.  $5797  k 

Paire  128. 

29.  :^,  A; 
h  B; 

A,  both ; 
4|  da. 

30.  3A  da. 


31 

.  30  da. 

32 

.  40,  shorter; 

84,  longer. 

33 

$43. 

34 

26t«7  da. 

35. 

45  ft.;  30  ft. 

36. 

00  and  80. 

37. 

$8269 k 

38. 

S2700,"  A  ; 

S3000,  B. 

39. 

S48. 

40. 

28  da.,  A; 

21  da.,  B. 

Paire  136. 

2 

.^aV- 

3.* 

31 

8  p. 

4. 

T5ff- 

5. 

^^' 

6. 

V^- 

7. 

"Nih- 

8-  juVo- 


A.NSU  KUS. 


379 


9.  I 

10.  m. 

11.  tL 

12.  j'A- 

\i  r  • 

15.  3^rt. 

3?:&. 

19.  4. 

20.    . 

21.    . 

22.  |. 

23.  i^. 

24.  A. 

25.  /;. 

26.  ,j, 

27.  rl 

28.  iJ^ 

29.  24 

S- 

30.  13 

If. 

Page  13 

2.  .25. 

3.  .6. 

4.  .625. 

5.  .375. 

6.  .8. 

7.  .75. 

8.  .0625. 

9.  .15. 

10.  .85. 

11.  .52, 

12.  .35. 

13.  .3833  4-. 

14.  ..555  +  . 

15.  .3846  4-. 

16.  .233  4-. 

17.  .1406  4-. 

18.  2.625. 

19.  .024. 

20.  .1875. 

21.  .0,3900  4-. 

22.  .42857  4-. 

2.$.  .4.545  4-. 

24.  .3157  4-  . 

25.  .23809  4- . 

26.  15.625. 

27.  24 

6. 

28.  .825. 

29.  3.425. 

30.  .23625. 

31.  .625. 

32.  .875. 

33.  .4375. 

34.  4.2155  4-. 

35.  37.54. 

36.  20.06. 

37.  .0001625. 

Page  138. 

2.  8.497. 

3.  8.7907. 

4.  8.914. 

5.  72.379. 

6.  .40035. 

7.  117.766. 

Pafre  139. 

8.  $108,455. 

9.  $68.19. 

10.  $5394.267. 

11.  30.975. 

12.  $32.1875. 

13.  .850955. 

14.  $77.7155. 

15.  $1102.345. 

16.  $48,365. 

Pa8:e  140. 

2.  24.903. 

3.  25.9()4. 

4.  $24,875. 

5.  $74,875. 

6.  .20695. 

7.  2.0232. 

8.  99.96154. 

9.  .0000756. 

10.  79999.92. 

11.  $11.94. 

12.  $17,705. 

13.  $4,105 

14.  $2519.98. 

15.  $76468.06. 

PaiCC  142. 

2.  .2210. 
8.  2.026. 


4.  .20056. 

5.  153.12. 

6.  2.2272. 

7.  25.752. 

8.  74.375. 

9.  .020265. 

10.  1822.5. 

11.  5.8776. 

12.  1.').1296. 

13.  :M.4576. 

14.  $23,375. 

15.  $167.48.5. 

16.  122.994. 

17.  28.0685. 

18.  $10.12. 

19.  2680.804. 

20.  1137.19424. 

21.  2.837025. 

22.  145.81944. 

23.  4.05U6288. 

24.  272.80767. 

25.  1.0725. 

27.  38464; 
3846.4 ; 
384640. 

28.  184.65; 
lvS46.5; 
18465. 

29.  $108,675. 

Pafco  e4:i. 

30.  $5212.81. 
;>1.  391.05  rd 

32.  $34631.37. 

33.  $60.6125. 

34.  $477,375. 

35.  $1706.16. 

36.  $71,445. 

Paire  140. 

2.  .25. 

3.  .005. 

4.  1.6895  4-. 

5.  .00365. 
"  C.  76.3. 

7.  30.2. 

8.  2.13. 

9.  .15. 
10.  3650. 


^5U 

ANSU  KiW. 

11.  27500. 

4.  r-'- 

10.  277i  lb. 

12.  2643.6923  +  . 

5.  1 

11.  1568.96  + lb. 

13.  .21. 

6.  2w-.   .   ;. 

12.  $292.25. 

14,  10020. 

7.  420200. 

13.  $437.r>0. 

15.  .81172-1-. 

8.  81000. 

14.  5200  lb. 

17.  .1 

9.  2 

15.  2000000. 

18.  .:. 

10.:.. 

16.  .00000002. 

19.  .01  :;>,:)  12. 

11.  $6.75. 

17.  25.818  + da. 

20.  .ois;>7. 

12.  $274,663. 

18.  $174.72.5. 

21.  .00006046. 

13.  $102. 

19.  102.56 +wk. 

22.  .384563. 

14.  $107,625. 

20.  $689,831. 

23.  3.5575 -f-T. 

15.  $246.25. 

21.  $14449.50. 

24.  180  doz. 

16.  $438,625. 

22.  $4,728. 

25.  $3.04. 

23.  $100000. 

2(5.  23  hhd. 

PflffO  ISO. 

24.  .2169  +  . 

27.  37  stove«. 

2.  $21.26|. 

25.  $.75^. 

3.  $40.89  -}- . 

26.  .36. 

Page  146. 

4.  $9.14  + . 

27.  241.96,    A ; 

2.  381744. 

5.  $7.>5.77J. 

120.975,  B: 

3.  4822336. 

6.  $356.85  +  . 

282.275,  C. 

4.  48518433. 

7.  $22.(Ui8  + . 

5.  8794128. 

8.  $11.06J. 

9.  $18,645. 

Page  101. 

6.  345684825. 

2.  006d. 

7.  47235605. 

10.  $72,875. 

3.  2658. 

8.  80952576. 

11.  $71.64.- 

4.  4148  far. 

9.  5862014256. 

12.  $.56.25. 

6.  90d. 

13.  $6:1085. 

7.  210d. 

Pafpe  147. 

14.  $264,158. 

8.  26^  far. 

10.  8:^73.45. 

11.  $1318.08. 

PmS«15S. 

9.  109Ad. 
10.  1068.; 

12.  $13483. 

2.  $293.06. 

5088  far. 

3.  $2935.00. 

11.  598  far. 

2.  240975. 

12.  1793d. 

3.  555489. 

Page  154. 

13.  843  far. 

4.  99949S. 

4.  $2393.60. 

14.  78.  6d. 

5.  1474812. 

5.  $26.20. 

15.  5436  far. 

0.  a511158. 

6.  $921.36. 

16.  8480d. 

7.  9398184. 

7.  $181.88. 

17.  43383  far. 

8.  148.50395. 

18.  28724  far. 

9.  1318044. 

Page  153. 

10.  1583232. 

1.  3631  lb. 

Page  163. 

11.  1564191. 

2.  127.'16975  lb. 

2.  78.  2td. 

12.  2415987. 

3.  20752?  fiq.  ft. 

3.  .£22  16s. 

13.  10L58948. 

4   36.153'+  cu.  ft. 

4.  £6  12s.  2d. 

14.  87855880. 

5.  2150420. 

5.  £4  Os.  6d. 

15.  107378577. 

0.  1000. 

7.  £Tio. 

7.  10000000. 

0.    y-  -778. 

Pagre  149. 

8.  261907.8. 

9.  £ri2s. 

3.  8600, 

9.  $16.36^. 

10.  £157  68. 

ANSWERS. 


381 


11.  XI 1  19s  Sd. 

12.  8(Jh  I'd. 

13.  X-242'll).s. 

14.  £0  48.  4d  . 

15.  JEIG  128. 

16.  978.  7 id. 

17.  £50  lis.  9^d. 

18.  3572  far. 

19.  £50  12s.  5d. 

20.  14784  far. 

21.  $72.9975. 

22.  £93  148.  Id. 

23.  £81  Is.  1(1. 

24.  £7  14s.  l^d. 

25.  $121,601. 

20.  14880  far. 

27.  £200. 

28.  £250. 

Pnyo  16G. 

13.  2  mi. 

14.  8903  vd. 

15.  1452  in. 
10.  443520  in.; 

570240  in. 

17.  19r\  rd. 

18.  180  rd. 

19.  121  mi. 
20  952204  in. 

21.  509716  in. 

22.  1  mi.  15  rd.  3  yd. 

1  ft.  4  in. 

23.  2  mi.  162rt!.  4  yd. 

2  ft. 

Pnico  107. 

24.  41775360  ft. 

25.  5  ft. 

26.  271  rd.  8  1. 

27.  47322  in. 

I>n«o  169. 

10.  12111  wi-in. 

11.  1 2044050860  sq.  in. 

12.  6  A.  4  Ri-  rd.  26 

sq.  vd.  2  M\.  ft. 
18.  14003316  8q.  in. 
14.  4  8q.  rd.  21  w\.  yd. 

1  ftq.  ft.  89  8q.  in. 


Pogro  I70. 

16.  88  sq.rd.  26  sq.yd. 

8  sq.  ft. 

17.  ^;i;J. 

18.  11  sq.  yd.  3  sq.  ft. 

13i  sq.  in. 

19.  77  sq'.  in. 

20.  120  8(1.  ft. 

21.  42  sq.  vd. 

22.  25  sq.  ft. 

23.  48  sq.  yd.; 
$55.20,  cost. 

24.  34  yd. 

25.  $79.80. 

26.  165  ft.  long; 
$35392.50,  cost. 

27.  80  rd. 

Vage  171. 

28   $472.50. 

29.  90  sq.  ft. 

30.  $35.20. 

31.  $48. 

32.  85J  sq.  vd.,  sides; 
34  sq.yd.,  ceiling; 
$44.23,  cast. 

33.  $21.39. 

Vnge  173. 

1.  3456;  5184; 
25920;  55296. 

2.  54;  81;  351;  675. 

3.  640;  1024. 

4.  16 J  perch; 
$29.56,  cost. 

5.  9/,. 

6.  207 if  yd. 

7.  55410  cu.  in. 

8.  369  cu.  ft. 

9.  S8i\  cu.  ft. 

10.  13824  blocks. 

11.  157  i  cu.  ft. 

Pnipc  174. 

12.  $10.31  i. 

13.  225  bu. 

14.  $348.44,  exeav'g; 
$418.18,  wall. 

15.  44517. 


1.  24  ft. 

2.  i;;'  It. 

3.  10"'  It. 


rage  175. 

5.  180  ft. 

-6.  $.37,312. 

-  7.  $53.90. 

8.  $27.56|. 

9.  $21.09|. 

Pnge  176. 

4.  296  pt. 

5.  $17.10; 
85 .\  gal. 

6.  120gal.  lpt.2gi. 
608gal.2(it.  Ipt. 

7.  135  gi. 

8.  4224  gi.;  60  gal. 

2  qt. 

9.  192  pt.;  8bbl.  12 

gal.  1  qt.  3  gi. 

10.  1617. 

11.  162?. 

12.  284    bbl.    30.623 

gal. 

Pafi^e  177. 

4.  251  pt. 
6.  525  pt. 

6.  526bn.l  pk.  5qt. 

PnKC  17H. 

7.  218  bu.  6  qt.  1  pt. 

8.  1792bu.2pk.4(it. 

9.  2301  pt. 

10.  15052.8;  17203.2; 
21504;  43008. 

11.  6.4388  4- ; 
16.6336  + ; 
22.6446  + . 

12.  483840  cu.  in.; 
225  bu. 

13.  260.357  4- . 

Pnse  179. 

4.  6205. 

5.  10216. 


383 


ANSWERS. 


C.  $10,875. 

7.  $2.52. 

8.  $42.24. 

PaV«  ISO. 

9.  100;  70;  98;  25. 
10.  $4.16.5. 
!1.  $3  25. 

12.  $9,885. 

13.  54.88  lb. 

14.  $54.32^. 

15.  $12. 

16.  982f. 

17.  54  bags. 

PafiTC  ISI. 

3.  3498. 

4.  7  oz.  4  pwt. 

5.  $74.25. 

C.  8  spoons;  1  oz.left. 

7.  564  powders. 

Ftige  l!i3. 

8.  18912. 

9.  23258. 

10.  13  hr.  31  min.  35 

sec 

11.  10  hr.  41  min.  37 

8CC. 

12.  2628000. 

13.  120  da. 

14.  197  da. 

15.  469  hr. 

16.  7815  min. 

17.  4  da.  10  hr.  50  min. 

18.  54738  sec. 

19.  3  wk.  1  da.  21  hr. 

25  mins. 

Vtkgc  185. 

2.  126000  sec; 
97200  sec; 
76338  sec. 

3.  123163  sec 

4.  130°  9'  20'^ 

5.  106°  48'  20^^ 

6.  648000;  324000; 
216000. 

7.  10800;  7200. 


Pa«e  1S6. 

2.  288;  492. 

3.  1728. 

4.  $16.20. 

5.  70  yr. 
0.  $9. 

P«ire  187. 

8.  7  02.  4  pwt. 

9.  8cwt  57  lb.  23  or. 

10.  26  rd.  3  vd.  2'ft. 

11.  43  sq.  ui.  19  8<i. 

yd.  2  8<i.  ft.  36 
sq.  in. 

12.  6qt.  1  pt.  1|  gi. 

13.  7  hr.  12  min. 

14.  18sq.yd.  8  sq.  ft. 

22jsq.in. 

15.  9cu.ft.777fcu.in. 

17.  H  pt. 

18.  f  ft. 

19.  I^sc. 

20.  .384  pt. 

Pa«e  188. 

22.  lis.  6d. 

23.  2  oz.  6  pwt.  10.56 

gr- 

24.  8  qr.  17.28  sh. 

25.  1  pt.  1.904  gi. 

26.  2  ft.  2.73  in. 

27.  145  rd.  9  ft.  10.8 

in. 

28.  1  pk.  7.4624  qt. 

29.  2  mi.  16  rd.  10  ft. 

6.72  in. 


jh  yd- 

jt\-^  hr. 


8. 
9. 

10.    j^hjT. 

12.  fs'^^cnJt, 

Page  189. 

13.  jjhj'B  '^' 

14.  j^^  bbl. 

6.  A. 

7.  if 


8.  i'l. 
9-  A- 

10.  ^i,s. 

Vnge  190. 

12.  .17708  + da. 

13.  .8125  bu. 

14.  .2121  -f-  rd. 

15.  mi. 

16.  .9239  +  £. 

17.  tVW  cwt 

18.  .1183  + mi. 

19.  .4597  -f  wk. 

20.  .4312  -I-  reams. 

21.  .4296  -f-  C. 

22.  .3717  -f  mi. 

23.  im  lb. 

1.  $4,805. 

2.  $2.34. 

3.  $19.68J. 

4.  $525.50. 

5.  21  lb.  4  oz. 

6.  $12,375. 

7.  lUC; 
$47.8l(. 

8.  7.01822  ft 

Pa«re  191. 

9.  $.1772+ per  oz.T. 

10.  2.6658+  A.; 
$399.87. 

11.  13f  da. 

12.  $1,728. 

13.  3200  mi. 

14.  1810. 

15.  $1.45  per  ream. 

16.  30  lb.  12.7  +  oz. 

17.  $113.13. 

18.  4248  posters; 
$27,612,  cost. 

19.  2.6277  bbl. 

20.  3  bu.  3  pk.  1  qt. 

1  pt.  too  much. 

21.  67500  lb. 

Pnfg:c  193. 

2.  991b.  loz.  15  pwt 

3.  £319  Os.  8^d. 


ANSWERS. 


383 


4.  37  mi.  121  nl.  1  v.l. 

1  ft.  1  in. 
r>.  25  T.  8  cwt.  40  lb. 

G  oz. 
0.  47  gal.  1  pt. 

7.  115  bu.  3pk.  4rit. 

8.  101  C.   125  cii.  ft. 

518  cu. in. 

10.  98  yd.  2  ft.  41  in. 

11.  2  A.  19  sq.  ril.  15 

sq.  yd.    1  sq.  ft. 
18  sq.  in. 

12.  109  gal.  1  qt.  1  pt. 

Page  194. 

13.  15  vr.  7  mo.  18  da. 

14.  22T.  17  cwt.  461b. 

2.  310  rd.  1  vd.  1  ft. 

8  in. 

Piige  105. 

3.  6  cwt.  95  lb.  2oz. 

4.  41  gal.  1  qt.  1  pt. 

3gi. 

5.  4  lb.  11  oz.  17  pwt. 
G.  8°  SO'  48'^ 

7.  18  C.  7  cd.  ft  15 

cu.  ft. 

6.  47  sq.  rd.  12  sq.  yd. 

Ssq.ft,  Hlsq.  in. 

10.  £8G  168.  8?d. 

11.  148  A.  147  sq.  rd. 

1 2  sq.  yd.  5  sq.ft. 
63  sq.  in. 

12.  6  R.  lOqr.  7sh. 

14.  Syr.  2  mo.  12 da. 

Vnge  196.  ' 

15.  66  yr.  8  mo.  13  da. 

16.  10  yr  7  mo.  20  da. 

17.  Feb.  13,  1841. 
18. 

19.  3vr.  11  mo.  28  da. 

20.  5  yr.  3  mo.  1  dx 

PlMr«  197. 

8.  69  gal.  3  (|t    1  l' 


;;.  159  lb.   1  oz.   12 
pwt.  15  gr. 

4.  12T.  3cwt.  531b. 

12  oz. 

5.  127  A.  15  sq.  rd. 
G.  116  rd.  1yd.  1  ft. 

7.  98  C.  3  cd.  ft.  8 

cu.  ft. 

8.  £32  198.  9d. 

9.  $49.95. 

Vngc  198. 

2.  2  bu.  7  qt.  1  pt. 

3.  85  A.  90  sq.  rd.  6 

sq.vd.  G4isq.in. 

4.  78  gal.  3  qt.  1  pt. 

2  gi. 

5.  1  T.  17  cwt. 

6.  2  C.  118|  cu.  ft. 

7.  £3  28.  6|d. 

9.  50  times. 

10.  4  da.  6  hr.  6  min. 

58  +  sec.' 

11.  23  mi.  24  rd.  10 

ft.  l|li  in. 

12.  272V5  bbl. 

13.  65  spoons;    1  oz. 

5  pwt.  left. 

14.  100  pickets. 

Pafce  201. 

3.  2  hr.  9  min.  13^ 

4.  122°  26'  45''  W. 

5.  5)ir.  5rain.  32sec. 

6.  5hr.8min. 

7.  11    min.    48  sec 

past  12  o'clock 
noon. 

8.  16°  17'. 

9.  53  min.    50  sec. 

A.  M.  Jan.  2. 
10.  17°  45'  east. 

P»C«906. 

8.  40.4671  +  A. 

9.  i  hectolitro. 

10.  5864.31  dm. 

1!.  ^71.:; I.'.. 


12.  $.257  + per.  gal. 

loss. 

13.  At  $3  per  metre; 
$.156  + per  yd. 

14.  15.96  M. 

15.  If  M. 

IG.  $46,067  + . 

17.  $:J500. 

18.  88.904  +  kilos. 

19.  18  hectares; 
$4500. 

20.  625  hectolitres. 

21.  $29,918. 

22.  11  cents  per  lb.; 
$11.34. 

23.  $10.05. 

rafi^c  £08. 

1.  A;  .10. 

2.  i;  .125. 

3.  I;  .20. 

4.  \]  .25. 

5.  VV;  .30. 

6.  J;  .75. 

7.  i;  .875. 

8.  J-,;  .03125. 

9.  ^;  .0625. 

10.  1};  1.25. 

11.  jh',  .015. 

12.  i;  .33k 

13.  i;  .16|. 

14.  2^;  2.125. 

15.  A;  .3125. 

16.  I;  .66^ 

17.  T»5;  .1875. 

18.  A;  .201. 

19.  X;  .0075. 

20.  ,L;  .005. 

21.  J5;  .046. 

22.  y§5u;  .003. 

23.  y^^;  .073. 

24.  jf;  .375. 

Pace  210. 

13.  $7.61. 

14.  $6.44. 

15.  155  gal. 

16.  306  mi. 

17.  300  sheep. 


384 

ANSWERS. 

ia  $300. 

29.  21 3  J. 

3.  $39,584. 

19.  $1293.36. 

4.  $5.38. 

20.  $.J02.G4. 

Pafire  216. 

6.  $3.96. 

21.  $.37500. 

9.  400. 

6.  $9.71. 

22.  $2625. 

10.  369. 

7.  $6.19. 

2.3.  $23200. 

11.  282. 

24.  S22940,  elder; 

12.  560. 

i»B«:e  223. 

$r2333.33J,y'nger. 

13.  $14000. 

8.  $8.61. 

14.  $300. 

9.  $8.95. 

Pnve  SIS. 

15.  $1500. 

10.  $7,917. 

16.  IGH. 

16.  $6000. 

11.  $6.56. 

17.  42H. 

18.  3lH%. 

19.  55  %, 

17.  800. 

Pairo224. 

Faue  217. 

2.  $1.5.22. 

20.  h\%. 

10.  525. 

3.  $76,826. 

21.  66H. 

11.  720. 

4.  $44.91. 

22.  38k. 

12.  633 J. 

6.  $18,835. 

23.  88  %  spent ; 
ll{%  left. 

Ji3.  426  ,V 

6.  $41. .54. 

14.  $1800. 

7.  $25.12. 

24.  33  % 

15.  500  bu. 

8.  $47,525. 

25.  H 

16.  $41.77i. 

—9.  ^6.03. 

26.  27^r%. 

17.  620  soldiers. 

10.  $38.61. 

27.  li^."^ 

18.  $50000. 

11.  $197.88. 

19.  $13400. 

12.  $92.3.5. 

Pare  213. 

13.  $66.29. 

28.  \\% 

Pa^e  221. 

14.  $39,807. 

29.  25%;  20%;  18  J%. 

14.  $2.26. 

15.  $131.29. 

15.  $5.92. 

16.  $1--'  '- 

Pagre  214. 

16.  $8.99. 

17.  r. 

9.  3080. 

17.  $645. 

18.  $a;>..  ... 

10.  2450. 

18.  $735.17. 

11.  833». 

19.  $773,256. 

Page  226. 

12.  21.45. 

20.  $979.87. 

2.  S49.55. 

13.  $532.50. 

21.  $737.76. 

3.  ^08.71. 

14.  343.75  bu. 

22.  $376^09. 

4.  $112.95. 

15.  2100  men. 

23.  $15.96. 

5.  $65.97. 

16.  3300. 

24.  $2.60. 

6.  $139.73. 

17.  3662. 

2.5.  $6.19. 

7.  $.54.61. 

18.  SI  5000. 

26.  $17,948. 

8.  $84.43. 

19.  $100000. 

27.  $5.55. 

9.  $64.89. 

20.  1102  A.  llOsq.rd. 

28.  $6.28. 

10.  $5.87. 

21.  $1600. 

29.  $8,788. 

11.  $368.96. 

22.  S18823-V 

30.  S17.89. 

23.  $8000. 

•  31.  $624,546. 

Page  228. 

24.  $20000. 

32.  $185.08,  interest. 

2.  $475.44. 

25.  $2700. 

$909.33,  amount. 

3.  $962.32. 

26.  $40206. 

27.  $20000000. 

Page  222. 

Page  229. 

28.  20. 

2.  $64,844. 

1.  $435.55. 

ANSWERS. 


n8/ 


2.  $402.88. 

3.  $91.4^). 

Page  331. 

2.  ^97.80. 

3.  $439,277. 

4.  iW  84 1.52. 

5.  $3980.35. 

6.  $219.31. 

7.  $15,086. 

PafTC  333. 

8.  $3022.56. 
0.  $<;2 1.056. 

10.  $833,818. 

4.  2  yr. 

5.  6  mo. 
0.  2  yr. 

7.  4  vr.  8  rao.  14  da. 

8.  5  yr.  4  mo.  13  da. 
0.  lOfyr. 

10.  12i  yr. 

11.  20  yr.;  16f  vr.; 
14^  vr. 

12.  40  yr.;  33 J  vr.; 
28f  yr. 


PAfire  334. 

5.  r.r^,. 

6.  7^/r. 


7.  5^%. 

8.  79^. 

9.  7^c. 

10.  7%. 

11.  6%. 

12.  79{,. 

13.  12J^. 

3.  S-- 

4.  > 

5.  $;........ 

6.  $1473.62. 

7.  $1,399.28. 

8.  $i:)71.43. 

9.  $10714.28. 

10.  $1125. 

11.  $.376,518. 

25 


12.  $297.26. 

Pasre  337. 

S802.75. 

$663. 

S2443.74. 

$3290.625. 

$1778.65. 


Page  339. 


2.  $894.95,     present 

worth ; 
$80.54  discount. 

3.  $777,195,  present 

wortli ; 
$68,005  discount. 

4.  $792,355,  present 

worth ; 
$166,395  disc't. 

5.  $464,717,  present 

worth ; 
$11 1.53,3  disc't. 

6.  $7698.254,    pres't 

worth ; 
$876,746  disc't. 

7.  $3948.26,  present 

wortli ; 
$325.73  disc't. 

8.  $2279.79,  present 

worth ; 
$565.20  disc't. 

9.  $1586.99,  present 

woBth ; 
$165.75  disc't. 

10.  $4489.07,  i)resent 

worth ; 
$1004.43  disc't. 

11.  $3306.30,  present 

worth ; 
$151.54  disc't. 

12.  $15547.169. 

13.  $28.44  gain. 

14.  $560.68. 

P»«<^  340. 


18.  $241.94. 


19.  $7441.018. 

20.  $1300.12  gain. 

Page  342. 

2.  $4,137. 

3.  $37,393. 

4.  $43,748. 
6.  $22.16. 

6.  $96.78. 

7.  $95.14. 

8.  $49.41. 

9.  $8.67. 

10.  $3,179  discount; 
S:i78.383  proc'ds. 

11.  $10.90  di.scount; 
$879.34  proceeds. 

12.  $3.94-f. 

1").  >1".^.91  discount ;~ 
,-rl")'>.S7.04pr(x;'d8. 

PnK<>  343. 

14.  $3729.79. 

15.  $563.80. 

Page  344. 

6.  $1000. 

7.  $2000. 

8.  $987.09. 

9.  $1015.74+. 

10.  $1408.1H-. 

11.  $1262.29. 

12.  $5299.46. 

13.  $1937.56. 

14.  $507.09. 

15.  $15316.54. 

l*agc  34ff. 

1.  $17500. 

2.  $140.81   in  favor 

of  49^  discount. 

3.  $8168.80. 
4..  $4137.93. 
6.  $4663.39. 

6.  1231.77. 

7.  $8.06+. 
s.  $1969.93. 

Page  346. 

9.  $1186.65. 


'Mii 

'  \-\\  I'l;-. 

10.  $267.07. 

iii.  4L"^7c,  A'again; 
'          50,VV^,B^«gain. 

16.  $3255. 

11.  $8800. 

17.  $3670. 

12.  $9422.22. 

1   32.  51A%. 

18.  8000  bu.; 

13.  27. 

$66.75  coram  is'n. 

14.  $9  per  bbl. 

1              P«C«SS8. 

19.  $1226. 

15.  $24.31. 

34.$l,co«t; 

20.  $1274.625. 

16.  $180.81. 

$1.10,  Bell'g  price. 

21.  2\Jc. 

17.  $iooooa 

35.  $9. 

36.  $1. 

37.  $.40. 

22.  $2776.19. 

P«««  S4«). 

PM«e25S. 

14.  $50. 

88.  $.55. 

1.  $9. 

15.  $18a 

39.  $100. 

2.  25%. 

40.  $300. 

3.  10%. 

P»S«2'll». 

41.  $2. 

4.  33il%. 

2.  $343.75. 

42.  $1.15. 

5.  125%. 

3.  $358.40. 

43.  $37472. 

6.  $3.10. 

4.  $397.8U. 

5.  $51.75. 

Pll««SSS. 

7.  $400. 

8.  $1.50. 

6.  $6.40. 

46.  $.40. 

9.  6%. 

7.  $96.88J. 

47.  $40000. 

10.  n  yr. 

8.  $6480. 

48.  $160869.5a 

11.  50%. 

9.  $3450. 

49.  $.10. 

12.  42f%. 

10.  $1605. 

50.  $5373. 

13.  Pay  cash;  l^,(^. 

11.  $272.50. 

51.  $.50. 

14.  28^%. 

12.  $2,795. 

52.  $2000. 

53.  $4.1 2 A-. 

VHge  259. 

Paffe2S0. 

54.  $3. 

15.  $40. 

13.  $74.75. 

55.  $1.55|. 

16.  6  bbl. 

14.  $422.15|. 

56.  $5.50. 

17.  50%. 

18.  Sell  now  at  20c.; 

15.  $4.80. 

57.  20%. 

16.  .325;  .65;   .975; 

by  $10.48+. 

1.1375. 

Fuse  2{IS. 

19.  $350,  cost  of  the 

18.  33}9<e. 

2.  $87.56. 

horse ;                     n 

19.  25%. 

3.  $180. 

$175,  cost  of  the 

20.  28^9^,. 

21.  20%. 

Pa«re  256. 

carriage. 
20.  $477.01. 

22.  25%, 

5.  $3932.03. 

21.  $15.30    gain     by 

23.  lli%. 

6.  14297.3+  yd. 

borrow'g  at  6%. 

24.  12ff9^,. 

7.  $50.05. 

22.  595.1  +  bu. 

8.  $39.2U. 

23.  $200. 

Pa^re  251. 

9.  $19,125. 

24.  $125,  A'scosi; 

25.  2lcf„ 

$100,  cost  to  me. 

26.  $2,296  per  box  ; 

Pagre  257. 

$.875  gain. 

10.  $8.91. 

Page  260. 

27.  11^9^,. 

11.  $25,088. 

25.  $20407.50. 

28.  eef^',.               ; 

12.  $7,084. 

26.  $25777.89. 

29.  12+%.                     1 

13.  $34,828. 

27.  $3288.38. 

30.  For  cash  at  once ;  j 

14.  $1495.09.                ' 

28.  Lost  $6;  4%. 

by  3,^%. 

15.  4557.03+ vd.         1 

29.  Neither. 

ANSWERS. 

387 

no.  is.oG-L  <;,, 

5.  $5418.75. 

Pti{(«  280. 

31.  S7:J.229  gain. 

6.  $288.75. 

2.  $187.50. 

32.  §120,  selling  price 

7.  $11407..50. 

3.  $67.50. 

of  horse ; 

4.  $270. 

5.  $1359.375,  prem.; 

$45,  selling   price 

Page  274. 

of  cow. 

9.  25  shares. 

$107390.625,  h^ss. 

l>afr«  261. 

10.  290  shares. 

6.  $515,125. 

11.  600  shares. 

7.  $55000. 

:;;;.  A,$200; 

12.  34.6  shares. 

8.  2%. 

B,  S133'; 

13.  22.11+  shares. 

•  9.  \f. 
10.  $14000. 

C,  $.53 J. 

15.  $209,234. 

34.  40  cents  per  Ih. 
3.3.  $4424.79+ 

10.  $622,707. 
17.  5%  at  60; 

12;$9&. 

m  9^9^. 

by  $10,666. 

13.  $2668.7.5,  los.s  of 

37.  $6, 

18.  $641.25. 

the  merchant ; 

38.  $4.50. 

19.  0%at90; 

$22331.25,  insur- 

30. $6267.49; 

by  $1.04. 

ance  co.'s  loss. 

20000  lb.  w<x>l. 

14.  $13333.335. 

40.  $1500. 

Pngrc  275. 

Page  261. 

20.  $109.62  dinain'd. 

Pagrc  2^1. 

21.  $1,%00. 

15.  $88400. 

■'■,  A  : 

22.  $15208.33. 

16.  $297000. 

.  .  .,  H: 

23.  $20812.50. 

17.  $6666.0()i  stock ; 

;rt>.i./-V  c. 

24.  $32770.04. 

$133.33.33},  store. 

25.  $12075. 

18.  $1.5228.42. 

Page  265. 

4.  $12,211  tax 

I»affe  276. 

Pllg«  'iS'Z, 

.003691  +  ruu . 

-7.  tiiVf- 

2.  $93.90. 

5.  .002272 -f. 

28.  lO^'c. 

3.  $145.09. 

6.  .007587+ rate; 

29.  O^^c  i^lock  at  \^(fc 

4.  $3699.50. 

$83,457  tax. 

discount. 

6.  $1423.45  loss. 

7.  $3826.53,    sum    to 

30.  N.  Y.  7'8;  /Aq^. 

6.  $1423..30  loss. 

be  asseflseil; 

31.  8,1,%.          ^^'^ 

7.  $465  less. 

$1168406+,  value 

33.  214?%. 

of  proper!  V. 

■'  ^     -'>'^c. 

Pag;^  2S6. 

Paffe  26(i. 

''h 

3.  $1005. 

4.  $3037.;"i0. 

2.  $178,125. 

5.  $4978.75. 

3.  $2.55. 

^.-i>. 

6.  $1471. 

4.  $450. 

3l>.  ^7S  12.03. 

7.  $4987.50. 

5.  $533.12. 

8.  $3003.75. 

(5.  $5248.246. 

Pace  277. 

9.  $4928.75. 

7.  $3322.512. 

40.  $4694.81. 

41.  $6888.3r).                , 

Pac«2>i7. 

Pac»373. 

42.  6|%. 

43.  \\\% 

10.  $1486.25. 

2.  $8593.75. 

11.  $4952..50. 

3.  $8670. 

44.  (Gained  $36.73.      i 

14.  $5710.72. 

4.  $1595. 

45.  Lo«  $26.19. 

15.  $1506.40. 

388 


ANSWERS. 


16.  $1213.04. 

PiilCe  300. 

3.  A,  ^'MO; 

2^ 

A  ^'^1(M)- 

B,  >>}'»; 

Pm^£ss. 

17.  $10012.51. 

B,  $2100; 

C,  $1800. 

4.  B,  >7n:,; 

C,  $7 10.25 ; 

D,  $1057.50. 

5.  A,  $426,505; 

18.  $3514.93. 

3. 

A,  $<)()0; 

19.  $1747.81. 

B,  $1200; 

C,  $800. 
A,$20(K); 

B,  $1600; 

C,  $2400. 
.\,$200; 
B,$160; 
C,  $280. 

A,  $600; 

B,  $750; 

C,  $675; 

D,  $975. 
D,$1600; 

2.  £571  48.  6W. 

3.  £732  6».  2|d. 

4.  £1149  68.  7W. 

5.  £1061  78.  5Jd. 

6.  $668.87. 

7.  $1832.25. 

8.  7021  .nr  fr. 

9.  :^' 

10.  ^ . 

11.  $3UG3.11. 

12.  $4484.11. 

4. 
5. 
6. 

7. 

B,  $621,987; 

C,  $426,505. 

6.  A,  $2550; 

B,  $3400; 

C,  $2550. 

7.  A,  $1080; 

B,  $1600; 

C,  $1820. 

8.  G,  $561,702; 
L,  $702,127; 
F,  $936,170. 

8. 

G,$2000; 
L,  $1800. 

E,  $862.50; 

F,  $575; 

Page  310. 

3.  S125. 

4.  $8.75. 

Pair<^  393. 

G,  $862.50. 

5.  $45. 

9. 

A,  $2744.78; 

6.  26i  T. 

7.  24fx. 

2.  2  mo.  29  Aa 

B,  $2299.:33 ; 

3.  2  mo.  18(1 

C,  S144().G3. 

8.  4aJ 

9.  12  men. 

4.  1  mo.  12  tUt. 

10. 

A,  $74.86; 

5.  2  mo.  10  da. 

B,  $86.84; 

10.  9^^^  da. 

6.  3^  mo. 

C,  $104.81; 

11.  36ttbu. 

Pai:e394. 

D,  $203.63. 

12.  1000  bbl. 

13.  40^  da. 

2.  June  .20,  1877. 

Pagre  301. 

3.  Mav  2,  1877. 

11. 

A,  $1200  Rain; 

Page  311. 

4.  Jan.  24,  1877. 

B,  SlGOOgain; 

14.  520  bu. 

5.  Dec.  22,  1876. 

C,  $7000  stock. 

15.  2H  A. 

0.  April  23, 1877. 

12. 

A,  $335,365; 

16.  $1628.25. 

7.  June  7,  1877. 

B,  $402.439 ; 

17.  65  da. 

8.  Aug.  16,  1877. 

C,  $536,585; 

18.  9792§  lb. 

Ph^e  SS96. 

D,  $670,731  ; 

E,  $804,878. 

19.  2307^  mi. 

20.  162^^.-  "ii- 

2.  Au?.  22,  1877. 

13 

A,  $750; 

21.  427?,  rd. 

3.  Mar.  21,  1877. 

B,  SI 000; 

22.  42  8  iir. 

C,  $1250. 

23.  20  da. 

Page  297. 

4.  June  19,  1877. 

Page  302. 

Page  313. 

5.  June  15,  1877. 

'       o 

.  A,  $2880; 

2.  28^  da. 

(>.  Julv  5,  1877. 

B,  S3600 ; 

3.  17820  lb. 

7.  $1198.60. 

1 

C,  $2B80. 

,      4.  $6875. 

5.  $710.76  + . 

ANSWERS. 

9.  821. 

Page  333. 

«.  lOjVrda- 

10.  886. 

3.  42. 

P«Ke  314. 

11.  969. 

12.  2424. 

4.  64. 

5.  65. 

7.  $.V.)4. 

13.  3546. 

6.  89. 

8.  $,<)02.77. 

14.  5555. 

7.  57. 

i).  473  vd. 

15.  472;  3375. 

8.  63. 

10.  .Slirini. 

10.  .874;  ..5555. 

9.  177. 

11.  1*")-;. 

10.  126. 

12.  2l\Ui. 

Pngre325. 

11.  536. 

13.  27000  lb. 

17.  .306. 

12.  1.259+. 

14.  18750  lb. 

18.  .315. 

13.  2.0800+. 

lo.  21  |T. 

19.  Iff. 

20.  III. 

14   .6463+  ;  .8617+ . 

10.  15|da. 

15   .8735-j  ;  .8434+; 

17.  546^5  ft. 

21.  m- 

^.\. 

Tnge  316. 

22.  .70710+. 

23.  .86602  4-. 

1.  45  ft. 

2.  1728;  12167; 

24.  .79056  +  . 

2.  24  in. 

59319;  13824. 
3.  2209;  2601; 

25.  .9486+. 

Pagrc  334. 

841;  1156. 

1.  25  ft. 

3.  13  ft. 

4.  22.5;  1089;  576; 

2.  45  rd. 

4.  10.75  ft 

1206;  625. 
5.  21952;  91125; 

3.  52  ft.  wide ; 
104  ft.  long. 

5.  10.81  ft. 

6.  21.50  ft. 

5832;  9261;  68921. 

4.  480  rd. 

7.  12.9+  in. 

7.  u't;  t^/s;  HI; 

5.  240  rd. 

8.  19.37  in. 

w^?m 

6.  ^4. 

9.  $19.51. 

^:^i^- 

Page  327. 

10.  3.46+  ft.,  width; 
10.38+ ft.,  length. 

10.  i')(;2."). 

2.  25  ft. 

11.  liectangle; 

11.  27U0U. 

3.  113.137  ft. 

143.55  +  sq.  ft.      • 

12.  .0000(M)25. 

4.  40  ft. 

12.  39.37  in. 

13.  .000000125. 

5.  122.474  ft. 

14.  4.2025. 

6.  140.584  mi. 

Page  3.15. 

15.  in. 

2.  10  ft. 

16.  m. 

Paifc  82*. 

3.  $3375. 

17.  9l|. 

7.  75  rd. 

4.  10.76  ft. 

18.  6415. 

8.  119.482  ft. 

5.  2.38. 

19.  9.0200V 

9.  205.704  ft. 

6.  Twice  as  great. 

20.  20.251285H. 

10.  172.046  ft. 

7.  27  times. 

21.  10000;  512;  729. 

11.  386.003  ft. 

8.  l8t,.506; 
2d,  .721 ; 

Pi«oS34. 

Pave  320. 

3d.  2.773. 

3.  53. 

2.  25.1328. 

9.  23.48. 

4.  63. 

4.  63.245. 

5.  66. 

5.  12  ft. 

PaKc3»8. 

6.  96. 

6.  100  rd.  length; 

2.  55. 

7.  2m. 

10  nl.  brcndth. 

3.  198. 

8.  344.  - 

7.  15.94268  rd. 

4.  $1.72. 

390 


AN.SUKilS. 


5.  209j'2  ft. 

6.  5070. 
8.  3775. 

0.  505. 

10.  3:^  mi. 

11.  78. 

12.  $3360. 

2.  2430. 

Pnce  340. 

3.  10240. 

4.  $984.15. 

5.  $133.82+. 
G.  $696,849. 
8.  1364. 

,!!•  !***• 

10.  4. 

1.  31.416  ft. 

2.  141.372  ft. 

3.  2  mi.  302.48  n\. 

4.  125.664  rd. 

Pa^e  344. 

5.  34.557  ft. 

6.  101.38  rd. 

7.  204.354  rd. 

1.  520  sq.  ft. 

2.  25|  sq.  ft. 

3.  720  sq.  rd. 

4.  525  sq.  ft. 

Pn^c  345. 

1.  216  sq.  ft. 

2.  126  sq.  ft. 

3.  S16S.376. 

4.  5935.85  sq.  ft. 

5.  $8.64. 

6.  15000  sq.  ft. 

rage  316. 

1.  5500  sq.  ft. 

2.  54  sq.  rd. 

3.  576  sq.  ft. 

4.  S546.875. 

5.  $2062.50. 


PRffe  347. 

1.  19.635  sq.  ft. 

2.  50.265  sq.  ft. 

3.  1145.9  sq.rd. 

4.  795  77  sq.  ft. 
6.  50.929  A. 

6.  706.86  s<i.  rd. 

7.  7.136  r.l. 

8.  12  rd. 

9.  2  acres  33.3939 

sq.  rd. 
10.  962.115  sq.  I 


Pnse  SOO. 

31.416  sq.  II 


2.  40  sq.  ft. 

3.  144  sq.  ft. 

4.  37.6992  sq.  ft. 

5.  63  sq.  ft. 


3S1. 

1.  540  sq.  ft. 

2.  376.992  sq.  ft. 

3.  628.32  sq.  ft. 

4.  $64. 

5.  89.5356  sq.  ft. 

6.  400  sq.  ft. 

7.  75.3984  sq.  ft. 

8.  157.08  sq.  ft. 

rnge  352. 

1.  251.328  sq.ft. 

2.  3000  sq.  ft. 

3.  $5,654. 

4.  340  sq.  ft. 

5.  $26.88. 

1.  4.908  sq.  ft. 

2.  1.396  sq.  ft. 

3.  26.50+  sq.  ill. 

Page  353. 

4.  45.830  sq.  ft. 

1.  2  en.  ft. 

2.  7.0686  cu.  ft. 

3.  $9. 

4.  462.857+  bn. 

5.  2632.089+  gal. 


n.  S  1013.837. 

l*nKe  354. 

1.  81.82;V2  cu.  ft. 

2.  18000  cu.  ft. 

3.  12440.736  lb. 

4.  7296  lb. 

1.  4666J  cu.  ft. 
Pafpe  355. 

•:.  236.405  4  on.  ft. 
:.  I.mi36+  cu.  ft 
1.  6415.718  gal. 

1.  65.45  cu.  ft.      , 

2.  268.0832  cu.  ft. 

3.  14.1372  cu.  ft. 

4.  795.217+  lb. 

5.  8.181  cu.  ft. 

6.  8181.25  cu.  ft. 

Vnge  350. 

1.  10  da. 

2.  $12. 

3.  80  men. 

4.  5§  mi. 

5.  16  men. 

6.  S5}. 

7.  S75. 

8.  336  lb. 

9.  $125,635. 

10.  13^  oz. 

11.  60  sheep. 

Pa»c  357. 

12.  $80. 

13. 5^^ 

14.  $40.33  loss. 

15.  451  trees. 

16.  $48,  A'sraonev; 
$40,  B's  money. 

17.  S65,  A's; 
$50,  B's; 
S55,  C's. 

18.  48 Jj  ft. 

19.  A's,  105  1b.; 
B's,  140  1b.; 
C's,  245-lb. 


ANSWEIIS. 

391 

20.  A'8,$32; 

Pfiffe  35». 

58.  B'sajje,  60  vr.; 

B'8;$27. 

:«.  72  n)i. 

C's  age,  80  yr. 

21.  34,    A'flage; 

34.  200. 

59.  $3000. 

46i,  B'8  age ; 

35.  57^. 

60.  $80. 

56|,  C's  age. 

36.  15. 

61.  2  cts.,  apples; 

37.  30  da. 

3  cts.,  pears. 

PaUrc  35S. 

38.  $?9. 

62.  A,46fda.; 

22,  720  apples. 

39.  15f.  yd. 

B,  35  da. 

23.  18ijg&  da.,  time 

in 

40.  50foz. 

63.  $1186.98. 

which  all  can 

do 

41.  566  t!le.«?. 

64.  S40  loss; 

it; 

42.  S2875.    • 

Wc.    . 

58 §1  da.,  time 
which  A  can 

in 

43.  26J. 

65.  27.64  ft. 

do 

44.  27^"*^  min.  past  5. 

66.  15  yr. 

it; 

45.  6  mi.  per  day. 

70U  da.,  time 
which  B  can 

in 

46.  30  men. 

Pngre  362. 

do 

67.  90  lb. 

it; 

Pnge  360. 

68.  162  in  No.  1 ; 

46/A  da.,  time 
which  C  can 

in 

47.  20  min.  past  5. 

144  in  No.  2; 

do 

48.  $882.46. 

128  in  No.  3. 

it. 

49.  $74.07. 

69.  A,  in  24  da.; 

24.  61  da. 

25.  $240,  one; 

50.  $740.52  4-. 

B,  inl7f  da.; 

51.  161%  gain. 

52.  42S%. 

C,  in  40  da. 

$150,  other. 

70.  A's8hare,$;i; 

26.  810  revolutions. 

63.  $270,  carriage ; 

B's  share,  $21. 

27.  72  ft. 

$240,  horse. 

71.  92160  A. 

28.  $3.80. 

54.  300  cats. 

72.  21. 

29.  8  hr.  48  min. 

55.  $24. 

73.  $600. 

30.  $208.33^. 

56.  $500,  cost. 

74.  $4563,  son's; 

31.  $1.50,  wheat; 

$1521,  widow's; 

$  .40,  corn. 

Vnge  361. 

$507,  daughter'a. 

32.  900  rd. 

57.  7  da. 

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